The paper concerns two interacting consumer-resource pairs based on chemostat-like equations under the assumption that the dynamics of the resource is considerably slower than that of the consumer. The presence of two different time scales enables to carry out a fairly complete analysis of the problem. This is done by treating consumers and resources in the coupled system as fast-scale and slow-scale variables, respectively, and subsequently considering developments in phase planes of these variables, fast and slow, as if they are independent. When uncoupled, each pair has unique asymptotically stable steady state and no self-sustained oscillatory behavior (although damped oscillations about the equilibrium are admitted). When the consumer-resource pairs are weakly coupled through direct reciprocal inhibition of consumers, the whole system exhibits self-sustained relaxation oscillations with a period that can be significantly longer than intrinsic relaxation time of either pair. It is shown that the model equations adequately describe locally linked consumer-resource systems of quite different nature: living populations under interspecific interference competition and lasers coupled via their cavity losses.
1. Introduction
Recently, there has been a great deal of activity aimed at studying the synchronization of coupled oscillators of diverse nature [1–5]. The theory of synchronization implies that even in uncoupled state the individual elementary units exhibit self-sustained oscillations. However, no less interesting are the systems where local coupling is essential for the very generation of oscillations and not only for their modulation or phase adjustment.
As far back as in early 1970s, Smale [6] constructed a counterintuitive mathematical example of a biological cell modeled by the chemical kinetics of four metabolites, x1,…,x4, such that the reaction equations dx/dt=R(x) for the set of metabolites, x=(x1,…,x4), had a globally stable equilibrium. The cell is “dead,” in that the concentrations of its metabolites always tend to the same fixed levels. When two such cells are coupled by linear diffusion terms of the form M(x2-x1), where M is a diagonal matrix with the elements μkδkl, however, the resulting equations are shown to have a globally stable limit cycle. The concentrations of the metabolites begin to oscillate, and the system becomes “alive.” In Smale’s words, “there is a paradoxical aspect to the example. One has two dead (mathematically dead) cells interacting by a diffusion process which has a tendency in itself to equalize the concentrations. Yet in interaction, a state continues to pulse indefinitely.”
The reaction equations involved in Smale’s model were too general to appeal to any specific process. Since then, inspired by his pioneer work, a number of models have been proposed containing biologically plausible mechanisms by which coupling of identical nonoscillating cells could generate synchronous oscillations. Among them are a model of electrically coupled cells, characterized by an excitable membrane and calcium dynamics [7]; a model in which coupling a passive diffusive cytoplasmic bulk with an excitable membrane (having an activator-inhibitor dynamics) produces a self-sustained oscillatory behavior [8]; an analog operational-amplifier implementation of neural cells connected by passive coupling (where conductance of the resistive connection simulates the diffusion coefficient) [9], and so forth. The authors of the last model suggested a term “awakening dynamics” for the phenomenon.
The subject of the present paper is an emergence of collective oscillations in a system of coupled nonoscillatory consumer-resource (CR) pairs. Owing to simplicity, this model in a sense may be considered minimal. Our choice of coupled CR equations as a matter of enquiry is dictated primarily by the ubiquity and importance of CR interactions.
CR models are the fundamental building blocks used in mathematical description and simulation of ecosystems. Depending on a specific nature of the involved CR interactions, they can take the forms of predator-prey, plant-herbivore, parasite-host, and victim-exploiter systems [10]. However applications of the CR models extend far beyond the ecology and are found wherever one can speak of win-loss interactions. In its broad meaning, resource is any substance which can lead to increased growth rate of the consumer as its availability in the environment is increased. As this takes place, the resource is certainly consumed. Consuming the resource means tending to reduce its availability. When carefully examined, CR models are identified in the following fields: epidemiology (susceptible and infected [11, ch. 10]), laser dynamics (photons and electrons [12, ch. 6]), labor economics (share of labor and employment rate [13, p. 28]), theoretical immunology (antigens and B lymphocytes [14, p. 299]), kinetics of chain chemical reactions (lipid molecules and free radicals [15]), and in numerous other studies from diverse disciplines.
We consider the situation when each of two consumer species exploits one respective resource only. As explained in the Appendix, terms “consumer” and “resource” in our model may bear not only their literal ecological meaning, but the physical meaning of photon density and population inversion in a laser cavity as well. Both resources are being supplied with constant rates like in a chemostat and consumed according to a simple mass-action kinetics. The resources are thought to be noninteractive. When uncoupled, self-inhibition of the individual consumer population is due to intraspecific interference. The coupling is assumed to originate solely from the interspecific interference competition between the consumers and quantitatively expressed by a bilinear term combining the competitor densities. Thus, the per individual loss rate of either consumer is proportional to the density of its counterpart.
Representation of competition between species in terms of loss-coupling dates back to the classical model of Lotka-Volterra-Gause (LVG) [16]. The LVG model operates with carrying capacities of the species, rather than referring explicitly to any essential resources. As shown by MacArthur [17], LVG equations may be considered as a quasi-steady-state approximation to the CR equations accentuating resource-mediated nature of competition, under the assumption of relatively rapid dynamics of the involved resources. Thereafter trophic competition have developed into a major descriptor of competition in the ecological literature generally and in conceptualizing ecosystems as systems of coupled CR oscillators specifically [5]. In contrast to the prevailing models of competition we consider the case of pure interspecific interference competition between the consumers with no consumption-induced contribution. Actually, our model is nothing more nor less than LVG equations augmented with the rate equations for the resources. Another key assumption of the model is that the dynamics of the consumers is much faster than that of the resources.
From a physical perspective, by and large similar equations with the like time hierarchy (fast consumer and slow resource) have been in use for coupled longitudinal modes in a semiconductor laser with an intracavity-doubling crystal since the work of Baer [18]. These equations have been treated mostly numerically. The notable analytical result belongs to Erneux and Mandel [19] who succeeded to show that the system admits antiphase periodic solutions by reducing it to the equations for quasi-conservative oscillator. However this result has to do with the onset of low-amplitude quasi-harmonic oscillations. Unlike their study, our approach deals with well-developed high-amplitude essentially nonlinear oscillations. Besides, we propose the model to be valid not only for competing laser modes, but for loss-coupled lasers as well.
We analyze the model using geometric singular perturbation technique according to which the full system of equations is decomposed into fast and slow subsystems. As we shall see below, the model reveals qualitatively different behavior at intense and weak competition between the consumer species. If coupling is strong, one of the consumers wins and completely dominates. When coupling is weak, the model exhibits low-frequency antiphase relaxation oscillations with each species alternatively taking the dominant role.
2. The Model
The two-consumer, two-resource model we consider is the following nondimensional system of four ordinary differential equations:(1a)u˙1=γ1-(u1+1)v1-u1,(1b)u˙2=γ2-(u2+1)v2-u2,(1c)εv˙1=(u1-δv1-ϰ2v2)v1,(1d)εv˙2=(u2-δv2-ϰ1v1)v2.Here dots indicate differentiation with respect to the nondimensional time variable t, ui and vi (i=1,2) are quantities measuring the respective population sizes of ith resource and ith consumer, γi>0 (i=1,2) is the inflow rate of ith resource, 0<δ≪1 is a parameter representing consumer self-limitation, ϰj>0 (j=1,2 and j≠i) quantifies the inhibitory effect of jth consumer on the growth of ith consumer due to coupling, and 0<ε≪1 is a singular perturbation parameter indicating that the dynamics of the consumers is much faster than that of the resources.
It should be mentioned that, being proportional to its dimensional prototype, vi directly represents population density of consumer species and is always nonnegative. Quantity ui, however, is not a population size in the true sense of the word. It is rather an affine transformation of a population size of the form N→aN+b, where a and b are scaling constants. This is done for reasons of mathematical convenience. Unlike a purely linear transformation, an affine map does not preserve the zero point, so in (1a)–(1d) ui=-1 corresponds to zero population size in reality. Nevertheless, from here on we shall apply the term “resource” to ui for brevity.
For more details and discussion on the derivation of the model (1a)–(1d) from different perspectives the reader is referred to the Appendix.
3. Model Analysis and Implications3.1. A Single CR Pair
When ϰ1,2=0, the communities are uncoupled and completely independent. An isolated CR pair is governed by equations:(2a)u˙=γ-(u+1)v-u,(2b)εv˙=(u-δv)v.
There exist two nonnegative steady states:(3a)u¯=γ,v¯=0;(3b)u¯=12(1+(4γ+2+δ)δ-1-δ)=γδ+O(δ2),v¯=12δ(1+(4γ+2+δ)δ-1-δ)=γ-γ(γ+1)δ+O(δ2).The linearization of (2a) and (2b) takes the form
(4)J=(-v¯-1-u¯-1v¯εu¯-2δv¯ε),
where (u¯, v¯) is one of the above steady states (3a) and (3b).
At (3a), one eigenvalue is negative and one is positive: λ1=-1, λ2=γ/ε. Thus (3a) is a saddle point. At (3b), TrJ=-(δ/ε+1)v¯-1<0 and detJ=(1+2δv¯+δ)v¯/ε>0, so the steady state is a stable node/focus. Specifically, focus is the case for
(5)(TrJ)2-4detJ=(γ2δ2+O(δ3))ε-2+(-4γ+O(δ))ε-1+O(1)<0,
whence one obtains an asymptotic estimate
(6)δ=o(ε1/2).
Figure 1(c) illustrates this focus-node bifurcation numerically. Damped oscillations is a well-known inherent feature of the photon-carrier dynamics in class-B lasers.
Phase portraits of an isolated CR system (2a) and (2b) for two cases when the unique stable equilibrium (3b) is a node (a), and a focus (b). Shown are vertical (u˙=0) and horizontal (v˙=0) nullclines. Steady states are marked by open circles. Numerical simulations have been carried out with data from Table 2. The respective values of the second order loss parameter δ in (a) and (b) are 0.1 and 0.01. Typically, consumer scale (v) is enormous; the equilibrium point happens to lie very close to the origin. For better appearance, henceforward we display all the graphs using coordinate transformation u→arsinh(u/γδ) and v→arsinh(v/γ). (c) Eigenvalues of (4) at (3b) as a function of the second order loss δ. Critical value δcr indicates a boundary between the regions of damped oscillations and aperiodic damping.
An isolated CR system (2a) and (2b) admits, therefore, the only solution which tends asymptotically towards the unique steady state. Periodic solutions are excluded. However the temporal dynamics of approaching this steady state essentially depends on the interplay between ε and δ, and is worth another look. There are in fact three timescales, O(ε), O(1), and O(δ), involved in the CR system (2a) and (2b) when it is overdamped, and two—O(ε) and O(1)—when underdamped.
System (2a) and (2b) is singularly perturbed because the derivative of one of its state variables, v, is multiplied by a small positive parameter ε. Singular perturbation cause two-time-scale behavior of the system characterized by the presence of slow and fast transients in the system’s response to external stimuli.
Replacing t in (2a) and (2b) with a fast time variable τ=t/ε and setting ε=0 we obtain the fast subsystem:(7a)u′=0,(7b)v′=(u-δv)v,where prime means differentiation with respect to τ. In the stretched timescale τ the slow resource variable u, according to (7a), is replaced by its initial value and reckoned as constant parameter. Equation (7b) is of a logistic type which has the solution
(8)v(τ)=u/δ1+(u/δv(0)-1)exp(-uτ),
valid for t=O(ε).
After a lapse of considerable time (in the fast scale) v converges to either of two fixed points depending on a sign of u:
(9)limτ→∞v(τ)={uδ,ifu>0;0,ifu<0.
It means that every single trajectory starting within the positive quadrant of (u,v) plane far enough from the stable steady state (3b) will run almost parallel to the vertical axis and hit the line u-δv=0 practically in a finite time of order O(ε). This is shown in Figures 1(a) and 1(b).
Now set ε=0 in (2a) and (2b) to get the slow subsystem:(10a)u˙=γ-(u+1)v-u,(10b)0=(u-δv)v.Equation (10b) describes a slow manifold consisting of two lines in the (u,v) plane: v=u/δ and v=0. By (9), the former attracts all the trajectories in the first quadrant, while the latter—all those in the second. In as much as the quasi-steady state of (7b), v¯=u/δ, is an isolated root of (10b) and v¯ is a stable solution of (10b) for any u>0; the assumptions of Tikhonov’s theorem [20] are satisfied, and one may proceed to approximate u and v in terms of the solution of the reduced system (10a) and (10b) in the slow timescale. It means that after arriving at the slow manifold v=u/δ, the representing point of the full system (2a) and (2b) will move along the manifold toward the equilibrium point with a characteristic velocity of order O(1).
In the immediate proximity to equilibrium (3b) the behavior of the trajectory is determined by the type of the fixed point, whether it is a node or a focus. Eventually this depends on the value of δ. Namely, close to a stable node, the system has two distinct real negative eigenvalues, one fast (λ1), and one slow (λ2):
(11)λ1=(-γδ+O(δ2))ε-1+(δ-1+O(1))+O(ε),λ2=(-δ-1+O(1))+O(ε),for1>δ≫ε.
Since |λ1|≫|λ2|, trajectories starting off the associated eigenvector 2 (which is tangent to the slow manifold v=u/δ) converge to that vector along lines almost parallel to eigenvector 1 (which is parallel to the vertical axis v). As they approach vector 2 they become tangent to it and move along it up to the very nodal point (Figure 1(a)). The characteristic time constant of this final stage is of order O(δ).
In the vicinity of a stable focus the motion is qualitatively different: trajectories still keep converging to the equilibrium, but no longer follow the slow manifold v=u/δ (Figure 1(b)). The reason is that for a focus eigenvectors are complex. Recalling that condition (6) must be true for a focus, calculate the eigenvalues in a limit case of δ→0:
(12)λ1,2=-12(γ+1)±iγεforδ=0.
Loosely speaking, one may think that near a focal point separation of state variables into slow and fast ones ceases to have its conventional meaning. There is no point to talk about motion along the slow manifold since there are no reduced one-dimensional systems corresponding to the neighborhood of a focus. The perturbation parameter ε affects only the frequency of damped oscillation, but not the damping rate. The radius of the focal spiral uniformly shrinks with a time constant of order O(1).
To O(1) for small δ there is a way to recast (2a) and (2b) near the focal equilibrium (0,γ) in a convenient form where the perturbation parameter ε does not multiply any right hand side. Namely, performing the scaling t=μ2s, u=μ2γξ, and v=γ(1+η), where μ2=ε/γ, one obtains
(13)ξ˙=-η-μ2ξ(1+γ(1+η)),η˙=ξ(1+η).
Here dots stand for differentiation with respect to the time variable s.
Equations (13) represent a weakly perturbed Hamiltonian system. For μ=0 the system is pure Hamiltonian and admits the first integral
(14)H=12ξ2+η-ln(1+η),
which is a conserved quantity (H˙=0). The periodic solutions of the Hamiltonian system form a one-parameter family with the equilibrium (0,0) as center point. The condition 0<μ≪1 makes (13) quasi-conservative with phase trajectories slowly spiralling to the equilibrium (Figure 2).
Phase-plane trajectory of the weakly perturbed Hamiltonian system (13) for γ=1.19375, ε=0.727273×10-3, and μ=0.157107. Near the equilibrium the oscillations are almost harmonic.
System (13) can be rewritten as a second order differential equation for η only:
(15)η¨+η+η2-η˙21+η+μ2(1+γ(1+η))η˙=0.
This is the equation for a nonlinear quasi-conservative oscillator. As is seen from (15), for |η|≪1 and |η˙|≪1 the oscillator is not only quasi-conservative, but also quasi-linear with the frequency ω0=1. These conditions motivate introducing the new variable ζ=η/μ. As a result one obtains
(16)ζ¨+ζ=μ(-ζ2+ζ˙21+μζ-μ(1+γ+μγζ)ζ˙).
This equation of quasi-conservative and quasi-linear oscillator will be used later on when analyzing the onset of synchronous periodicity.
3.2. Coupled Communities: Stability
In the following, we will distinguish two main types of coupling: strong, ϰ1,2>1, and weak, ϰ1,2<1.
Physically feasible equilibria, or fixed points, (u¯1,u¯2,v¯1,v¯2), of (1a)–(1d) are those for which v¯1,2⩾0. We denote the interior fixed point by F12=(u¯1,u¯2,v¯1,v¯2), where the subscripts stand for the consumers. Lack of a certain index at a boundary fixed point means that the consumer concerned is not present (extinct). Thus F1=(u¯1,u¯2,v¯1,0) and F2=(u¯1,u¯2,0,v¯2) designate either of one-consumer equilibria corresponding to dominance, while F=(u¯1,u¯2,0,0) means both consumers having been washed out.
Model (1a)–(1d) has four feasible steady states. To O(1) for small δ(17a)F:u¯1=γ1,u¯2=γ2,v¯1=0,v¯2=0;(17b)F1:u¯1=0,u¯2=γ2,v¯1=γ1,v¯2=0;(17c)F2:u¯1=γ1,u¯2=0,v¯1=0,v¯2=γ2;F12:u¯1=ϰ1γ1-ϰ2γ2-ϰ1ϰ2+1±R2(ϰ1-1),u¯2=-ϰ1γ1+ϰ2γ2-ϰ1ϰ2+1±R2(ϰ2-1),(17d)v¯1=u¯2ϰ1,v¯2=u¯1ϰ2,where
(18)R=(ϰ1γ1-ϰ2γ2-ϰ1ϰ2+1)2+4ϰ2(ϰ1-1)(ϰ1γ1-γ2).
Boundary equilibria F, F1, and F2 always exist. F12 exists if
(19)1ϰ2<γ2γ1<ϰ1forϰ1>1∧ϰ2>1(strongcoupling),
or
(20)ϰ1<γ2γ1<1ϰ2forϰ1<1∧ϰ2<1(weakcoupling).
In case (19) the expression with “+” in (17d) is realized, while in case (20) the one with “−” is feasible.
F is always unstable, because two of the four associated eigenvalues are positive:
(21)λ(F):γ1ε,γ2ε,-1,-1.
Correct to O(1) in ε, the eigenvalues for F1 and F2 are(22a)λ(F1):-1,-12(γ1+1)±iγ1ε,γ2-ϰ1γ1ε;(22b)λ(F2):1,-12(γ2+1)±iγ2ε,γ1-ϰ2γ2ε,whence it follows that (17b) and (17c) are stable when(23a)γ2γ1<ϰ1,(23b)γ1γ2<ϰ2,respectively.
The necessary and sufficient conditions for all the eigenvalues of the Jacobian matrix,
(24)(-v¯1-10-ϰ2v¯2-100-v¯2-10-ϰ1v¯1-1v¯1ε00-ϰ2v¯1ε0v¯2ε-ϰ1v¯2ε0),
evaluated at F12, to have negative real parts are, from the Routh-Hurwitz criterion, (25a)c0>0,(25b)c3>0,(25c)c2c3-c1>0,(25d)c1(c2c3-c1)-c0c32>0,where c0, c1, c2, and c3 are the coefficients of the characteristic polynomial λ4+c3λ3+c2λ2+c1λ+c0 of (24):
(26)c0=v¯1v¯2(ϰ1(v¯1-ϰ2(v¯1+v¯2+1))+ϰ2v¯2+1)ε-2,1c1=-ϰ1ϰ2v¯1v¯2(v¯1+v¯2+2)ε-2+(v¯1(v¯2(ϰ1+ϰ2+ϰ1v¯1+ϰ2v¯2+2)+1)+v¯2)ε-1,c2=-ϰ1ϰ2v¯1v¯2ε-2+(v¯1(v¯2(ϰ1+ϰ2)+1)+v¯2)ε-1+(v¯1+1)(v¯2+1),c3=v¯1+v¯2+2.11111111111111111111111111111
To analyze the validity of (25a)–(25d) we put ϰ2=μϰ1, where μ=O(1). Then for (25a) to be true, ϰ1=o(1) should be met, which is incompatible with strong coupling, yet possible for weak coupling. Conditions (25b) and (25c) are always valid. As is known, (25d) guarantees a simple complex conjugate pair of eigenvalues corresponding to a linearization about steady state F12 to have negative real part. For ϰ1=o(1) it boils down to
(27)(-v¯1v¯2(v¯1+v¯2+2)(v¯1+v¯2+v¯12+v¯22)μϰ12+O(ϰ13))ε-3+((v¯1+1)(v¯2+1)(v¯1-v¯2)2+O(ϰ1))ε-2+O(ε-1)>0,
whence it follows that
(28)ϰ1=o(ε1/2).
With regard to a fairly small value of ε, (28) may be thought to be broken under most physically meaningful conditions unless coupling is infinitesimally weak. Hence, normally, condition (25d) of the Routh-Hurwitz criterion is never fulfilled and the interior fixed point F12—if it exists—is always unstable by growing oscillations.
Existence and stability conditions of possible nonnegative equilibrium points are summarized in Table 1.
Existence and stability conditions of nonnegative equilibria of system (1a)–(1d).
Equilibrium
Existence
Stability
F
Always
Unstable
F1
Always
γ2/γ1<ϰ1
F2
Always
γ2/γ1>1/ϰ2
F12
1/ϰ2<γ2/γ1<ϰ1 for ϰ1,2>1 (strong coupling)
Unstable
ϰ1<γ2/γ1<1/ϰ2 for ϰ1,2<1 (weak coupling)
ϰ1,2=o(ε1/2)
3.3. Strong Coupling: Bistability and Hysteresis
Strong coupling in (1a)–(1d) makes bistability of boundary equilibria possible, as evident from Table 1. When both F1 and F2 are stable with an unstable coexistence steady state F12, the system being studied is able to exhibit a hysteresis effect. Given the strong coupling, suppose that the inflow of resource 2 is kept at some constant level, γ2=γ2*, while the inflow of resource 1, γ1, steadily increases from a value less than γ2*/ϰ1 along the line γ2=γ2* in the (γ1,γ2) parameter plane, as shown in Figure 3(a). Referring to (17a)–(17d) and (23a) and (23b), one sees that initially F1 is unstable; F2 is stable and F12 does not exist with the complete dominance of consumer 2. Within the interval γ2*/ϰ1<γ1<ϰ2γ2* steady state F1 becomes stable yet empty and F12 becomes existent (according to (19)) yet unstable, so the situation remains unchanged until point (ϰ2γ2*,γ2*) in Figure 3(a) has been reached from the left. For a larger γ1, state F2 gives up its stability and the system jumps to F1. Consumer 2 gets washed out, while consumer 1 takes over. If now we start reducing γ1, the system remains in F1 until γ1 drops to the lower critical value γ2*/ϰ1, beyond which F1 is no longer stable and there is a reverse jump to F2. In other words, as γ1 progresses along γ2=γ2* there is a discontinuous switch from consumer 2 to consumer 1 at ϰ2γ2*, while as γ1 retraces its steps, there is a discontinuous switch from consumer 1 to consumer 2 at γ2*/ϰ1. Figure 3(b) illustrates how the steady-state level of consumer 1 responds to infinitesimally slow changes of the inflow rate of its own resource. The hysteresis is made possible thanks to the concurrent stability of both boundary equilibria and instability of the interior fixed point for γ1∈(γ2*/ϰ1,ϰ2γ2*). In terms of electronics, such a situation would describe a flip-flop circuit—bistable multivibrator—having two stable conditions, each corresponding to one of two alternative input signals.
Hysteresis in the two-consumer, two-resource system (1a)–(1d). It takes place in the case of strong coupling ϰ1,2>1. (a) The resource-supply parametric plane (γ1,γ2). Consumer 1 completely dominates below the line γ1-ϰ2γ2=0, whereas consumer 2—above the line ϰ1γ1-γ2=0. Both boundary steady states are stable in the region of bistability confined by the two aforementioned lines and marked off by gray, with the dominance of either consumer being a matter of path-dependency. (b) Equilibrium level of consumer 1 as a function of the resource supply. Numerical values of the coupling strengths are chosen to be ϰ1=1.2 and ϰ2=1.5.
As seen from Table 1, the very existence of interior equilibrium F12 in a case of weak coupling (condition (20)) implies instability of both boundary fixed points, F1 and F2. System (1a)–(1d) happens to possess four nonnegative steady states, none of them being stable. As we have found, F12 is unstable through growing oscillations. In such a case, the model would thus be expected to have a limit cycle in its four-dimensional phase space corresponding to self-sustained oscillations.
As is known, Hopf bifurcations come in both super- and subcritical types. If a small, attracting limit cycle appears immediately after the fixed point goes unstable, and if its amplitude shrinks back to zero as the parameter is reversed, the bifurcation is supercritical; otherwise, it’s probably subcritical, in which case the nearest attractor might be far from the fixed point, and the system may exhibit hysteresis as the parameter is reversed.
To check whether the bifurcation is supercritical or subcritical, we employ the averaging method [21]. For negligible δ the problem reduces to weakly coupled quasi-conservative oscillators. Introduce the new time s=t/μ2, where μ2=ε/γ1. The new dynamic variables ξ1,2 and η1,2 are defined by formulas ξ1,2=u1,2/μ2γ1,2 and η1,2=(v1,2-γ1,2)/γ1,2. Thus the variables ξ1,2 and η1,2 are the respective deviations of u1,2 and v1,2 from their standalone steady-state values. With these new variables equations (1a)–(1d) become(29a)ξ˙1=-η1-μ2ξ1(1+γ1(1+η1)),(29b)η˙1=ξ1(1+η1)-(γ2γ1)ρ2(1+η1)(1+η2),(29c)ξ˙2=-η2-μ2ξ2(1+γ2(1+η2)),(29d)η˙2=(γ2γ1)ξ2(1+η2)-ρ1(1+η1)(1+η2),where dots mean differentiation with respect to s and ρ1,2=ϰ1,2/μ2 are the scaled coupling strengths.
For brevity, we restrict our consideration to the case of identical CR pairs and symmetric coupling setting γ1,2=γ and ρ1,2=ρ. With these assumptions system (29a)–(29d) can be transformed to two coupled second-order equations for quasi-conservative, quasi-linear oscillators. This can be done as follows. First, (29b) and (29d) are solved for ξ1 and ξ2. Secondly, (29b) and (29d) are differentiated with respect to time to get η¨1 and η¨2. Thirdly, ξ1, ξ2, ξ˙1, and ξ˙2 are plugged into the equations for η¨1 and η¨2. And finally, η1,2 are replaced by the new variables ζ1,2: η1,2=μζ1,2.
As a consequence, we arrive at the following coupled equations:
(30)ζ¨1,2+ζ1,2=-ρζ˙2,1+μZ1(1,2)+μ2Z2(1,2)+O(μ3),
where
(31)Z1(1,2)=-ρ(γ+1)-ζ1,22+ζ˙1,22-ρζ1,2ζ˙2,1,Z2(1,2)=-(γ+1)(ζ˙1,2+ρζ2,1)-ζ1(ρ(2γ+1)+ζ˙1,22).
We seek the solution of (30) in the quasi-harmonic form
(32)ζ1,2=a1,2cos(ωs+φ1,2),ζ˙1,2=-a1,2ωsin(ωs+φ1,2),
where a1,2 and φ1,2 are slowly varying amplitudes and phases, and ω is the frequency of the synchronous oscillations. Differentiating the assumed form of ζ1,2 and equating the result to the assumed form of ζ˙1,2 yields the first pair of relationships between a1,2 and φ1,2:
(33)a˙1,2cos(ωs+φ1,2)-a1,2φ˙1,2sin(ωs+φ1,2)=0.
Then, differentiating ζ˙1,2 and substituting the resulting expression for ζ¨1,2 as well as the assumed forms for ζ1,2 and ζ˙1,2 into (30) yields the second pair of equations relating a1,2 and φ1,2. Separating into equations for the rate of change of a1,2 and φ1,2 one obtains
(34)a˙1,2=ω-1sin(ωs+φ1,2)(A0(1,2)+μA1(1,2)+μ2A2(1,2)),φ˙1,2=a1,2-1ω-1cos(ωs+φ1,2)(A0(1,2)+μA1(1,2)+μ2A2(1,2)),
where
(35)A0(1,2)=-a1,2(ω2-1)cos(ωs+φ1,2)-ρa2,1ωsin(ωs+φ2,1),A1(1,2)=a1,2(-a1,2ω2sin2(ωs+φ1,2)+a1,2cos2(ωs+φ1,2)-ρa2,1ωcos(ωs+φ1,2)sin(ωs+φ2,1)-a1,2ω2sin2)+ρ(γ+1),A2(1,2)=a1,2cos(ωs+φ1,2)×(a1,22ω2sin2(ωs+φ1,2)+ρ(2γ+1))+(γ+1)×(ρa2,1cos(ωs+φ2,1)-a1,2ωsin(ωs+φ1,2)).
To this point no approximations have been made except of the expansion in powers of μ in (30). Averaging equations (34) over the period 2π/ω and considering a1,2, φ1,2, a˙1,2 and φ˙1,2 to be constants while performing the averaging, one obtains the following equations describing the slow variations of a1,2 and φ1,2: (36a)a˙1,2=12π∫02π/ω(A0(1,2)+μA1(1,2)+μ2A2(1,2))×sin(ωs+φ1,2)ds=-ρa2,1ωcosϕ+μ2(γ+1)(a1,2ω-ρa2,1sinϕ)2ω,(36b)φ˙1,2=12πa1,2∫02π/ω(A0(1,2)+μA1(1,2)+μ2A2(1,2))×cos(ωs+φ1,2)ds=ρ(a1,22+a2,12)sinϕ2a1,2a2,1+O(μ2),where ϕ=φ1,2-φ2,1.
Any stable fixed points of (36a) and (36b) could mean that the phase difference between the coupled oscillators do not change in time (ϕ=const), and the oscillations are periodic with constant amplitudes a1,2. Thus, finding the conditions when these fixed points are stable would mean finding the conditions at which the synchronization occurs. Equations (36a) and (36b) show that the first approximation of the averaging method does not predict any nonzero steady-state amplitudes. The reason for this seems to lie in the fact that the even terms in (35) do not contribute to the value of integral (36a). To take account of them we have to employ in what follows the second approximation of the averaging method. As to the phase difference, ϕ, we see that stable is only antiphase steady-state regime with ϕ=π.
Let R be the vector of the right-hand sides of (34). Also denote the operation of averaging by the angle brackets 〈·〉. Then 〈R〉 will mean the right-hand sides of (36a) and (36b). Following the conventional procedure we retain only vibrational terms in R,
(37)R~=R-〈R〉,
and integrate R~ up to an arbitrary function of the amplitudes chosen for simplicity to be equal to zero:
(38)R^=∫R~ds.
Now, upon calculation of the Jacobian matrix ∂R/∂(a1,a2,φ1,φ2)T, we can write down the second approximation symbolically as
(39)(a˙1,a˙2,φ˙1,φ˙2)T=〈R〉+〈∂R∂(a1,a2,φ1,φ2)TR^〉.
(The terms now neglected by averaging are of a higher order of magnitude with respect to the small parameters than the terms neglected in the first approximation.)
Due to the symmetry of the case we may set a1=a2=a enabling us to present the resulting equations of the second approximation as compact as possible: (40a)a˙=-12aρcosϕ+μ224ω3a(ρ(ωcosϕ(a2(2ω2+5)+12ρ(γ+1))111111111111+2ω2sinϕ(3a2ρcosϕ-ρa2+6(γ+1))111111111111-2a2ω(ω2+1)cos2ϕ111111111111-6(γ+1)sinϕω3a2(2ω2+5))-12(γ+1)ω3a2(2ω2+5)),(40b)ϕ˙=ρsinϕ+μ224ω2ρsinϕ(3a2(3ω2-10)+8a2(ω2+1)cosϕ111111111111111-24ρ(γ+1)(ω2+1)).
The stable steady-state solution for the phase difference is ϕ=π. Inserting this value into (40a) and putting a˙=0 we get three different steady-state solutions for a (including the trivial one), of which only one,
(41)a=23ρω2-μ2(γ+1)(ρ2+ω2)μρ(4ω2+7),
being stable. Formula (41) indicates a soft transition to sustained oscillations as the coupling parameter ρ is progressively increased from a critical value. Thus we expect that the Hopf bifurcation is supercritical.
Figure 4 plots (u1,v1) phase planes for coupling strength below and above the Hopf bifurcation. When ϰ1,2=0 (consumers are decoupled) the internal equilibrium is a stable focus (Figure 4(a)). For ϰ1,2=0.011 (very weak coupling; just below the bifurcation) F12 is still a stable focus, though a very gently winding one: the decay is slow (Figure 4(b)). For ϰ1,2=0.012 (very weak coupling; just above the bifurcation) there is an unstable focus at F12 and a stable oval limit cycle of small size representing low-amplitude quasi-harmonic oscillations (Figure 4(c)). On further increasing of coupling strength the limit cycle continuously grows in size and takes an irregular shape indicating nonlinearity of the synchronous oscillations (Figures 4(d)–4(f)).
The development of a limit cycle in model (1a)–(1d) in relation to the coupling strength. The projections of phase trajectories and fixed points on (u1,v1) plane are presented. “•” and “∘” mark the respective stable and unstable internal steady state F12; “×” stands for the boundary steady state F1. (a) ϰ1,2=0: CR pairs are decoupled; each has unique stable steady state. (b) ϰ1,2=0.011: very weak coupling; the spiral winds only algebraically fast. The system is on the verge of the Hopf bifurcation. (c) ϰ1,2=0.012: very weak coupling; the stable limit cycle is just born. (d) ϰ1,2=0.025: weak coupling; nonlinear oscillations. (e) ϰ1,2=0.05: moderate coupling; the limit cycle passes near the basin of the saddle point. (f) ϰ1,2=0.6: moderate coupling; the limit cycle is about to merge with a heteroclinic cycle.
As a practical matter, the range of very weak coupling not too far away from the Hopf bifurcation, where oscillations are quasi-linear and quasi-harmonic, is of less concern to us than is the range of far more feasible not-too-weak coupling, corresponding to well-developed substantially nonlinear oscillations. We are going to demonstrate that given conditions (20), system (1a)–(1d) exhibits relaxation oscillatory behavior, with the two coupled CR pairs being antiphase locked.
By the assumption, 0<ε≪1, meaning that system (1a)–(1d) is singularly perturbed. The slow variables are resources, u1 and u2, and the fast variables are consumers, v1 and v2. The standard practice of reducing such systems is the adiabatic elimination of the fast variables, when the left-hand side in the fast equation is replaced by zero, thus turning this differential equation into an algebraic equation. It is assumed that the fast variables quickly relax to their momentary equilibrium, quasi-steady-state, values obtained from the algebraic equations, in which the slow variables are treated as parameters. “Frozen” slow variables do not move substantially in this short adaptation time of the fast variables. Quasi-steady-state values of the fast variables can then be expressed by values of the slow variables. The fast variables hastily adapt to the motion of the slow variables. The former are entrained by the latter. The utility of quasi-steady-state approximation is that it allows us to reduce the dimension of the system by retaining only slow variables in the model. One has to establish the validity of the adiabatic elimination in each specific case by using the recommendations of the singular perturbation theory [22]. In particular, Tikhonov’s theorem [20] requires the quasi-steady state of the fast equations to be stable.
To decompose the full system (1a)–(1d) into fast and slow subsystems, introduce fast time variable τ=t/ε. Now rescale (1a)–(1d) by replacing t with τε and, after taking ε=0, it becomes(42a)u1′=u2′=0,(42b)v1′=(u1-δv1-ϰ2v2)v1,(42c)v2′=(u2-δv2-ϰ1v1)v2,where prime means differentiation with respect to τ. This is the fast subsystem, where u1 and u2 are replaced by their initial values and treated as parameters. It yields inner solution, valid for t=O(ε).
Setting ε=0 in (1a)–(1d) leads to the slow subsystem(43a)u˙1=γ1-(u1+1)v1-u1,(43b)u˙2=γ2-(u2+1)v2-u2,(43c)0=(u1-δv1-ϰ2v2)v1,(43d)0=(u2-δv2-ϰ1v1)v2,which produces outer solution, valid for t=O(1). In this singular limit as ε→0, the subsystem defines a slow flow on the surface (slow manifold) given by (43c) and (43d). Outer solution is valid for those u1 and u2, for which the quasi-steady states of the fast subsystem are stable.
We anticipate the dynamics of the full system (1a)–(1d) in its four-dimensional phase space (u1,u2,v1,v2) to consist of two typical motions: quickly approaching the slow manifold (43c) and (43d), and slowly sliding over it until a leave point (where the solution disappears) is reached. After that, the representing point may possibly jump to another local solution of (43c) and (43d).
Thus, we ought to find all quasi-steady states of the fast subsystem (42a)–(42c), map the domains of their stability onto the slow phase plane (u1,u2), and then investigate the dynamics of the slow subsystem (43a)–(43d) with piecewise continuous functions.
Fast subsystem (42a)–(42c), which is nothing but the conventional LVG model, has four quasisteady-states—three boundary and one interior—denoted by Q (the slow variables are deemed to be frozen):(44a)Q:v~1=0,v~2=0;(44b)Q1:v~1=u1δ,v~2=0;(44c)Q2:v~1=0,v~2=u2δ;(44d)Q12:v~1=ϰ2u2-δu1ϰ1ϰ2-δ2,v~2=ϰ1u1-δu2ϰ1ϰ2-δ2.Existence and stability of these quasi-steady states are determined by (u1,u2)—position of a representing point in the phase plane of the slow subsystem, shown in Figure 5(a).
(a) Phase plane of the slow subsystem (43a)–(43d) sectored (by dashed lines) into stability domains of the corresponding quasi-steady states of the fast subsystem (42a)–(42c). Both boundary quasi-steady states, each corresponding to the situation when either of the two consumers completely dominates, are stable within the opening of the angle formed by the dashed lines. Lines u1=γ1δ and u2=γ2 are the respective nullclines u˙1=0 and u˙2=0 of the piecewise subsystem (47a)-(47b). Lines u1=γ1 and u2=γ2δ mean the same for the piecewise subsystem (52a)-(52b). Intersections of the nullclines (marked by the open circles) are equilibria of the associated piecewise slow subsystems, and they must lie outside the above-mentioned opening to allow for the relaxation oscillations. (b), (c), and (d) are the respective phase portraits of the fast subsystem (42a)–(42c) generated by points “b,” “c,” and “d” in the slow phase plane (a).
Q, Q1 and Q2 always exist for all u1 and u2 from the positive quadrant of the slow phase plane. For not-too-weak coupling, such that ϰ1ϰ2>δ2, Q12 exists for all u1 and u2 satisfying the condition δu1/ϰ2<u2<u1ϰ1/δ, that is, within the opening of the angle formed by lines δu1-ϰ2u2=0 and ϰ1u1-δu2=0 in Figure 5(a). The opening shrinks as the coupling strengths get weaker.
Jacobian matrix of the fast subsystem,
(45)(u1-2δv~1-ϰ2v~2ε0-ϰ2v~1ε-ϰ1v~2εu2-2δv~2-ϰ1v~1ε),
has the following sets of eigenvalues at (44a)–(44d):(46a)λ(Q):u1ε,u2ε;(46b)λ(Q1):-ϰ1u1εδ+u2ε,-u1ε;(46c)λ(Q2):-ϰ2u2εδ+u1ε,-u2ε;(46d)λ(Q12):±u1u2+O(δ)ε.Based on (46a)–(46d) one concludes that Q (the origin) is always an unstable node for all u1 and u2 from the positive quadrant of the slow phase plane. Q1 is a stable node for δu2<ϰ1u1, that is, below the line ϰ1u1-δu2=0 in Figure 5(a), otherwise it is a saddle. Similarly, Q2 is a stable node for δu1<ϰ2u2, that is, above the line δu1-ϰ2u2=0 in the plane of slow variables, otherwise it is a saddle. The interior quasi-steady state, Q12, is always a saddle.
The performed typology of fixed points of the fast subsystem (42a)–(42c) leads to three qualitatively different phase portraits depicted by Figures 5(b)–5(d).
Suppose initially Q1 is stable and Q2 is not. Consumer 1 completely dominates. This corresponds to slow variables u1 and u2 being somewhere below the line δu1-ϰ2u2=0 of Figure 5(a). Fast subsystem (42a)–(42c) has phase portrait of a type shown in Figure 5(b). While u1 remains much greater than γ1δ, the dynamics of the resources (treated as bifurcation parameters in reference to the consumers) is described by a system of two independent equations:(47a)u˙1=γ1-(u1+1δ+1)u1,(47b)u˙2=γ2-u2,which is a piecewise version of the slow subsystem (43a)–(43d) for v1=u1/δ and v2=0. System (47a) and (47b) has stable steady state
(48)u^1(1)=12(r1-δ-1)=γ1δ+O(δ2),u^2(1)=γ2,
where r1=1+δ(4γ1+2+δ). This equilibrium lies in the upper left corner of Figure 5(a).
While heading to (48), the trajectory crosses the line δu1-ϰ2u2=0 and enters the domain of bistability of both Q1 and Q2. Fast subsystem (42a)–(42c) takes new phase portrait of a type presented by Figure 5(c). However the dominance of consumer 1 persists.
Upon introducing the deviations ξ1 and η2 from the respective steady states (48), the system (47a)-(47b) become(49a)ξ˙1=-ξ1(r1+ξ1)δ,(49b)η˙2=-η2,whence one finds
(50)ξ1(t)=r1(r1/ξ1(0)+1)exp(r1t/δ)-1,η2(t)=η2(0)exp(-t).
It follows from (50), that the dynamics of variable u1 is faster than that of u2 due to small δ. Clearly, the representing point must have relaxed to the vertical line u1=γ1δ well before approaching the horizontal line u2=γ2. Further developments depend on whether the involved individual CR systems are overdamped or underdamped.
If δ is great enough to damp intrinsic oscillations in the constituent CR pairs, the representing point will slide along the nullcline u˙1=0 (slow manifold of the system (47a)-(47b)), steadily tending to point (48) (Figure 6(a)).
The limit cycle of system (1a)–(1d) projected onto phase plane of the slow variables in the cases of (a) strong second-order damping, and (b) underdamping. The parameters of the model chosen for numerical simulation are ε=0.727273×10-3, γ1=1.19375, γ2=1, ϰ1=0.5, ϰ2=0.8, δ=0.1 (a), and δ=0.01 (b).
If δ is small in terms of the condition (6), then the individual CR pairs are underdamped. In the immediate vicinity of u^1(1) the division of variables on slow and fast ones loses its meaning and reduced (47a) is no longer valid. Instead of (47a) one should write down two equations:(51a)u˙1=γ1-(u1+1)v1-u1,(51b)εv˙1=(u1-δv1)v1.However this system is identical to (3a) and (3b) for an uncoupled CR system and, in essence, describes convergence to a focal point via dying oscillations. In the plane of slow variables (u1,u2) these oscillations manifest themselves in damped transverse fluctuations superimposed on the independent vertical motion along the nullcline u˙1=0 (u1=u^1(1)≈γ1δ) toward point (48) (Figure 6(b)).
By virtue of the condition (20), the trajectory has to cross the line ϰ1u1-δu2=0 on its way toward the neighborhood of steady state (48). As soon as this has happened, node Q1 in the plane (v1,v2) will be absorbed by saddle Q12. A new phase portrait of the fast subsystem (42a)–(42c) takes on the appearance of Figure 5(d). Consumer 1 rapidly washes out, and the alternative boundary quasi-steady state Q2 becomes stable, with consumer 2 dominating.
In terms of the four-dimensional phase space of full system (1a)–(1d), the representing point is now in the other stable branch of the slow manifold (43c) and (43d). The motion over this alternative branch obeys the piecewise subsystem:(52a)u˙1=γ1-u1,(52b)u˙2=γ2-(u2+1δ+1)u2,with the initial conditions u1(0)=u^1(1)≈γ1δ and u2(0)=γ1ϰ1.
The dynamics of (52a) and (52b) is basically similar to that of (47a) and (47b) analyzed above. System (52a) and (52b) has a stable steady state:
(53)u^1(2)=γ1,u^2(2)=12(r2-δ-1)=γ2δ+O(δ2),
where r2=1+δ(4γ2+2+δ). This equilibrium lies in the lower right corner of Figure 5(a).
Variable u2, being more rapid in comparison to u1, quickly enters the neighborhood of the nullcline u˙2=0 given by u2=u^2(2)≈γ2δ and then—depending on the value of δ—finally approaches the nullcline either monotonically (Figure 6(a)) or via damped oscillations according to equations(54a)u˙2=γ2-(u2+1)v2-u2,(54b)εv˙2=(u2-δv2)v2.(Figure 6(b)). System (54a)-(54b) describes underdamped intrinsic oscillations of uncoupled consumer 2 for small δ.
At the same time, u1 steadily and independently tends to u^1(2)=γ1. Again, because point (53) is located below the line δu1-ϰ2u2=0 (on the strengths of the condition (20)), the trajectory would certainly cross that line at a point (γ2ϰ2,γ2δ), whereupon node Q2 would be absorbed by saddle Q12. The system returns to the first branch of the slow manifold, and thereby the oscillatory cycle gets closed.
4. Results and Discussion
Figure 7 shows the results of a numerical integration of system (1a)–(1d) for the case of underdamped individual consumer-research pairs. The two coupled communities execute self-sustained relaxation oscillations which are antiphase-locked.
Time profiles of antiphase relaxation oscillations in two coupled CR pairs modeled by (1a)–(1d). (a) and (b) show the, respective, resource 1 and consumer 1; (c) and (d) display the, respective, resource 2 and consumer 2. Numerical values of the parameters are those mentioned in the caption of Figure 6(b).
The resources u1 and u2 demonstrate sawtooth periodic pulses. The oscillation range for the resource levels remains finite and, what is important, it does not depend on the intrinsic second order loss δ (measuring intraspecific interference).
The times of motion over either branch of the slow manifold (43c) and (43d) add up to give a predominant contribution to the period of oscillations, T. These times are determined mainly by the dynamics of the slow resource variables u1 and u2 and, to a zeroth approximation in ε and δ, can be found as solutions to the equations of motion (52a) and (47b) with respective boundary conditions (0,γ2ϰ2) and (0,γ1ϰ1). In this way one obtains a quite simple estimate for the period:
(55)T=∫0γ2ϰ2dzγ1-z+∫0γ1ϰ1dzγ2-z=ln11-ϰ2(γ2/γ1)+ln11-ϰ1(γ1/γ2).
It is interesting that, according to (55), the period depends on the ratio of the two resource inflows, γ1 and γ2, rather than on each of them individually, and does not depend completely on concrete value of δ.
The consumers v1 and v2 change periodically between extinction and respective constant levels γ1 and γ2. Very brief transient from zero to flat nonzero level within each cycle is accompanied by a highly pronounced spiky overshoot. The magnitude of the spike tends to infinity as δ→0, in view of (44b) and (44c). Depending on the intensity of intraspecific interference, the overshoot may or may not be followed by a tail of fading high-frequency oscillations, when a consumer variable falls below its steady-state value and then bounces back above, taking some time to settle close to its steady-state value. In signal processing, such a kind of transient oscillation is known as “ringing.” There is no ringing if the involved CR pair does not oscillate due to significant intraspecific interference. Ringing takes place if intraspecific interference is negligible and therefore the involved CR pair is characterized by underdamped intrinsic oscillations. The “pitch” of ringing is just the frequency of these intrinsic oscillations.
One notices that when one consumer is very scanty, the coupled system behaves like an isolated CR pair (2a) and (2b). Another essential feature of the dynamics is the role of the resource variables in determining when the consumers emerge and wash out. When u1, for example, rises above a threshold value (determined by the amount of losses experienced by v1), then v1 comes into dominance causing v2 in turn to disappear. So except for their transient spiking and ringing, the consumer levels, either flat nonzero or essentially zero, are determined by the hysteretic cycling of the respective resources.
Scrutinizing a cycle of consumer oscillations one may distinguish four parts within it:
v1 is essentially zero, while v2 is approximately equal to its uncoupled steady-state value, γ2. u1 increases due to its inflow until it overcomes losses for consumer 1;
with a sufficient resource stock, v1 now emerges. The population exhibits a spike due to the fast time scale of the consumer equations. The sharp increase in population saturates the available resource level, so u1 drops. Cross-losses cause v2 to wash out;
v1 and u1 relax to quasi-steady-state values, as if there were only one isolated CR pair. v2 is essentially zero. u2 is increasing, like u1 did in part 1;
u2 surpasses the losses, v2 emerges and the subsequent cross-losses cause v1 to wash out. The spiking v2 also causes a substantial decrease in the available stock of the associated resource. The sequence begins again.
Presenting his famous model Smale remarked that “it is more difficult to reduce the number of chemicals to two or even three” [6]. As distinct from Smale’s example, coupling in our case makes self-sustained synchronous oscillations possible for just two variables.
As we have seen, phase trajectory of the system constantly moves from the neighborhood of unstable boundary equilibrium F1, where only consumer 1 is present, to the neighborhood of F2, where consumer 2 completely dominates, back to F1, and so on in cyclic alternation. This kind of trajectory was termed “heteroclinic cycle” by Kirlinger [23]. A heteroclinic cycle occurs when the outflow (unstable manifold) from one saddle point is directly connected to the inflow (stable manifold) of another saddle point, and vice versa. It is closely related to another notion of the nonlinear dynamics, a homoclinic cycle, which emerges when the unstable and the stable manifolds of the same saddle coincide and form a closed loop.
Homo- and heteroclinic cycles are not robust structures in the sense that infinitesimally small change of system parameters destroy them. However in the practical sense, any limit cycle passing in close proximity to saddle points will be indistinguishable from a heteroclinic cycle (Figure 8). The only difference is strict periodicity, although the period of the limit cycle in a neighborhood of the heteroclinic cycle may be long. Besides, at the threshold of homo-/heteroclinic bifurcation the period is susceptible to external noise.
A 3D-projection of the limit cycle in system (1a)–(1d) for parameters chosen in a neighborhood of the heteroclinic cycle. See caption to Figure 6(b) for the parameters.
In the context of our model, as coupling becomes stronger, the stable limit cycle swells and passes closer and closer to boundary fixed points which are node-saddles or focus-saddles (Figures 4(d)–4(f)). Depending on the interplay between the parameters, eventually it may bang into one or both of these equilibria creating either a homoclinic or heteroclinic cycle, respectively. This corresponds to γ2/γ1=ϰ1 and γ1/γ2=ϰ2. On further increasing the coupling, the saddle connection breaks and the loop is destroyed.
It is worth noting that heteroclinic cycles were first found by May and Leonard [24] in a classical LVG system with competing three species. However in their model the cycle is not truly periodic: as time goes on, the system tends to stay in the neighborhood of any one boundary equilibrium ever longer, so that the “total time spent in completing one cycle is likewise proportional to the length of time the system has been running.” Moreover, May and Leonard state that “the phenomenon clearly requires at least three competitors, which is why it cannot occur in models with two competitors.” This statement is echoed by Vandermeer [25] who extended their theory on higher dimensions: “It appears to be the case that all cases of an odd number of species follow this basic pattern, whereas all cases of even number of species result in extinction of half of the components, leaving the other half living independently at their carrying capacities.” In view of our results, the above conclusion is by far and away true providing one stays within the framework of classical LVG equations, which in fact imply a high rapidity of the resource dynamics. In our model of just two competitors the slowness of the resource relative to the consumer is essential for the oscillations to occur, because it provides the necessary inertia to the system.
Physically, our model is most likely feasible because it is based on the well-established rate equations (A.8b) for semiconductor lasers, and therefore should be considered as a model of antiphase synchronization of two lasers via their loss-coupling.
Ecologically, the feasibility of the model is tightly bound to justification of the adopted time hierarchy in system (1a)–(1d). Time scales are usually inverted in ecosystems, as opposed to lasers, the most common case being rapid consumption of food by species. However it seems reasonable to propose that our model may describe the first level of an ecosystem, at which the consumers are autotrophs and the resources are mineral nutrients. The ability to exploit different substrates leads to a possibility of stable coexistence of different organisms descending from the common ancestor. Divergent evolution is just the formation of new species: due to mutations two populations emerge with the same genetic code but having proteins able to process different substrates. Providing the environmental conditions are quite stable on the evolutionary timescale, the inflows of inorganic substrates from the surroundings may be considered constant and the washout time of a substrate may occur much longer than the life expectancy of a species (recall the definition of ε from (A.6)).
5. Conclusion
We proposed a model of two CR pairs linked by interspecific interference competition. When uncoupled, an individual CR pair has a unique stable steady state and does not admit periodic solutions. If intraspecific interference within the species is strong enough, the equilibrium is nonoscillatory (stable node), otherwise the steadying occurs by decaying oscillations (stable focus).
When coupled, the model behaves differently at strong and weak competitive interaction between the consumers. When coupling is strong, one of the consumers wins. Which consumer wins or loses depends critically on the relative intensities of the resource inflows and coupling strengths. In the case of bistability, when the system acts like a bistable multivibrator (flip-flop circuit), the winner may be determined by the initial conditions. Any static coexistence of competing consumers is not possible.
When coupling is moderately weak, the model reveals low-frequency antiphase relaxation oscillations represented by a continuous flow of rectangular pulses. The system works as an astable multivibrator continually switching between its two states, neither of which is stable. The consumers cannot coexist even dynamically: in each of two alternating states one consumer completely dominates and the other is on the verge of extinction. The most intriguing feature of the model is that each of the participating CR pairs taken separately does not oscillate; both communities are completely quiescent, however, in interaction, when coupled in a nonlinear way, the resulting system turns into a relaxation oscillator.
One way or the other, it is believed that the proposed model for coupling-induced oscillations in nonoscillatory CR pairs can be considered as a minimal in that class of population-dynamical systems and its mechanism can be applied to networks with large numbers of nonoscillatory elements and complex architecture.
AppendixA. Derivation of the Coupled CR EquationsA.1. Ecological Perspective
Of all types of interactions between individuals of the same population (intraspecific interactions) or individuals of different populations (interspecific interactions) of the same trophic level competition is most commonly encountered. In a broad sense, competition takes place when each species (individual) has an inhibiting effect on the growth of the other species (individual). An inhibiting effect should be understood to mean either an increase in the death rate or a decrease in the birth rate.
Consider the famous CR equations proposed by MacArthur [17, 26]: (A.1a)x˙j=(rj(1-xjKj)-∑i=1ncijyi)xj,j=1,…,m,(A.1b)y˙i=(∑j=1mcijwjxj-bi)yi,i=1,…,n.Here, dots indicate differentiation with respect to time t, xj represents the total biomass of jth resource (prey), yi stands for the total biomass of ith consumer (predator) species, the constant rj defines the growth rate of jth resource, Kj is the carrying capacity of jth resource, cij is the rate of uptake of a unit of jth resource by each individual of ith consumer population, wj-1 is the conversion efficiency parameter representing an amount of jth resource an individual of ith consumer population must consume in order to produce a single new individual of that species, and bi is the loss rate of ith consumer due to either natural death or emigration. All parameters in (A.1a) and (A.1b) are nonnegative.
MacArthur assumed population dynamics of the resources to be much faster than that of the consumers which enabled him to approximate xj in (A.1b) by its quasi-steady-state value derived by setting the right-hand side of (A.1a) to zero. As a result, he succeeded to reduce slow-scale equation (A.1b) to the well-known LVG model [16]:
(A.2)y˙i=(ki-∑s=1naisys)yi,i=1,…,n,
where(A.3a)ais=∑j=1mcijcsj(wjKjrj),i=1,…,n;s=1,…,n,(A.3b)ki=∑j=1mcijwjKj-bi,i=1,…,n.
More recently, such an asymptotic reduction has also been carried out for a model of competition where species (with continuous trait) consume the common resource that is constantly supplied, under the assumption of very fast dynamics for the supply of the resource and fast dynamics for death and uptake rates [27].
CR model (A.1a)-(A.1b) assume that competition within and between consumer species is purely exploitative: individuals and populations interact through utilizing (or occupying) common resource that is in short supply. Quite on the contrary, LVG model (A.2), (A.3a)-(A.3b) describes competition strictly phenomenologically, as direct interference where consumers experience harm attributed to their mutual presence in a habitat (e.g., through aggressive behavior). However we have to stress that MacArthur’s reduction neither claims that interference competition entirely results from “more fundamental” trophic competition, nor urges us to hastily consider direct competition as some derived concept. What it actually states is that at slow-time scale associated with dynamics of the consumers, the effects of exploitation competition are indistinguishable from those of interference competition. And at slow-time scale, coefficients ais of (A.3a) merely add to interference coefficients ais′ which are to be present primordially in (A.1b).
Most mathematical models dealing with coupled CR pairs or multilevel trophic chains ignore contributions of intraspecific and interspecific interference. Indeed, the empirical data like [28] do indicate that aij′ may be negligible in comparison with ais. Still, works advocating the explicit account for direct interference show that incorporation of self-limitation and cross-limitation terms in the equations at the consumers’ level can provide for the stable coexistence of many species on few resources [29, p. 31], [30].
Moreover, if we are to assume dynamics of the resources to be much slower than that of the consumers, it is likely that we have to retain interference competition terms in all equations (A.1b).
Consider the following modification of (A.1a) and (A.1b) representing coupled two-consumer, two-resource equations:(A.4a)x˙1=p1-(c1y1+q1)x1,(A.4b)x˙2=p2-(c2y2+q2)x2,(A.4c)y˙1=(c1w1x1-b1-d1y1-h2y2)y1,(A.4d)y˙2=(c2w2x2-b2-d2y2-h1y1)y2.
Instead of the logistic mode of resource supply, as is the case in MacArthur’s model, our model is based on so-called “equable” mode of resource exploitation [31], by which the quantities of available resources are held constant by a continuous-flow system. According to (A.4a) and (A.4b), a constant concentration of jth resource (j=1,2) flows into a defined volume with the rate pj while unused resource flows out with the per capita rate qj, in much the same manner as in a chemostat [32].
In more exact terms, the true chemostat model for one substrate and one species looks as follows:
(A.5)x˙=D(x0-x)-μxyKx+x,y˙=(wμxKx+x-D)y,
where the rate of substrate uptake is expressed by the Monod formula μxy/(Kx+x), Kx is a saturation constant numerically equal to the substrate concentration at which the uptake rate is half the maximum, D is the dilution rate defined as the rate of flow of medium over the volume of the bioreactor, and x0 is an input concentration of the substrate.
Model (A.5) turns into an uncoupled version of (A.4a)–(A.4d) if we put p=Dx0 and q=b=D and assume Kx≫x, so that μx/(Kx+x)≈cx, where c=μ/Kx.
In natural conditions, the equable modes of feeding, for instance, can be found on the first trophic level of ecosystem, among autotrophs.
Besides, in (A.4c) and (A.4d) intraspecific competition strength di (i=1,2) measures direct interference of individuals within ith consumer population with each other resulting in an additional per capita loss rate diyi; interspecific competition strength hs (s=1,2; s≠i) quantifies direct interference effect from sth consumer on ith consumer resulting in an additional per capita loss rate, hsys, of the latter.
Equations (A.4a)–(A.4d) contain two important assumptions. First, they assume that the resources are noninteractive. On higher trophic levels, however, resources may interact and the possibility of competition among the resources was originally pointed out by Lynch [33]. Since then, a whole series of theoretical papers has been published on two-predator, two-prey systems with interference competition between two self-reproducing prey species based on the Lotka-Volterra equations. Specifically, Kirlinger [23] describes the model in which each predator specializes on one prey only, while Xiang and Song [34] treat a similar model in which each predator is allowed to feed on both prey.
As seen from (A.4a) and (A.4b), there is no intraspecific interference competition within the resource populations either. Yet the resource level would remain finite even in the absence of the consumer.
A second assumption of our equations is that the consumers interact only directly, through interference competition and cannot compete through their use of resources, as each consumer specializes in one resource only. The theory of pure trophic competition in equable models has been developed in the works [31, 35].
Intraspecific interference competition is allowed within the consumers as well. Even though the available amount of any resource happened to be of a constant level, the population size of the associated consumer would remain finite due to self-limitation caused by direct intraspecific interference.
The novelty of model (A.4a)–(A.4d) is that it considers time hierarchy of MacArthur’s CR equations to be reversed by assuming dynamics of the consumers to be much faster than that of the involved resources and articulates the importance of direct competition mechanisms within the framework of this assumption.
Upon the scaling
(A.6)u1=c1w1x1b1-1,u2=c2w2x2b2-1,v1=c1y1q1,v2=c2y2q2,γ1=c1p1w1b1q1-1,γ2=c2p2w2b2q2-1,δ1=d1q1b1c1,δ2=d2q2b2c2,ϰ1=h1q1b2c1,ϰ2=h2q2b1c2,β=q1q2,ε1=q1b1,ε2=q2b2,t′=q1t
equations (A.4a)–(A.4d) take the following nondimensional form:(A.7a)u˙1=γ1-u1v1-u1-v1,(A.7b)βu˙2=γ2-u2v2-u2-v2,(A.7c)ε1v˙1=(u1-δ1v1-ϰ2v2)v1,(A.7d)ε2v˙2=(u2-δ2v2-ϰ1v1)v2.Note that in (A.7a)–(A.7d) dots mean differentiation with respect to nondimensional “slow” timescale variable t′, as defined by (A.6).
The parameters β-1, ε1-1, and ε2-1 reflect the rapidity of the dynamics of u2, v1, and v2 with reference to that of u1. It is assumed that β=O(1) and ε1,2≪1.
In studying the effect of coupling, the parameters of interest are obviously the coupling strengths, ϰ1 and ϰ2. The parameters of interest are also those which characterize the difference between the states of the uncoupled systems. The resource income rates γ1 and γ2 are used as the control parameters that distinguish the relative base states of the two systems.
For the sake of simplicity but without any loss of generality, we set β=1, ε1=ε2=ε, and δ1=δ2=δ and also drop the prime at t, to obtain (1a)–(1d).
A.2. Laser Dynamics Perspective
Laser rate equations originally proposed by Statz and deMars [36] are differential equations that relate two quantities: injected carrier density (n) and photon density (p). For a single-mode semiconductor laser, these equations take the form [12, ch. 6]:(A.8a)n˙=Jqd-Γvga(n-n0)p-nτe,(A.8b)p˙=Γvga(n-n0)p-pτp, where dots mean differentiation with respect to time t, J is the injection current density (pump parameter), q is the magnitude of the electron charge, d is the active-layer thickness, Γ is the confinement factor accounting for the fraction of the light power contained in the active region, vg is the group velocity of light that can be expressed through the speed of light in vacuum (c) and the group refractive index of the dispersive semiconductor material (μg) as vg=c/μg, a is the gain coefficient, n0 is the carrier density at transparency corresponding to the onset of population inversion, τe is the lifetime of the electrons in the conduction band before being lost by escape from the active region, and τp is the lifetime of photons inside the cavity before going out of the cavity or being absorbed inside the cavity. In (A.8b) the contribution of spontaneous emission is neglected.
Typical parameter values for a semiconductor laser (mostly borrowed from [12, p. 238]) are given in Table 2. These numerical values are used in the calculations throughout the present paper, unless otherwise noted.
Laser parameters used for numerical simulations. The values in parentheses stand for associated nondimensional quantities.
Quantity
Notation
Meaning
Value
J (γ)
Pump current density
5×103 A/cm^{2} (1.19375)
d
Active layer thickness
2×10-5 cm
Γ
Confinement factor
0.3
μg
Group refraction index
4
a
Differential gain coefficient
2.5×10-16 cm^{2}
n0
Carrier density at transparency
1×1018 cm^{−3}
τe
Carrier lifetime
2.2×10-9 s
τp
Photon loss time
1.6×10-12 s
ε
τp/τe
0.727273×10-3
D (δ)
Intrinsic second order loss
0.773×10-5 cm^{3}/s (0.01)
Consider two (not necessarily identical) lasers of type (A.8a)-(A.8b) and introduce additional intensity-dependent losses such that each laser, i, of the two has a total loss represented by the sum of the constant loss, 1/τpi, plus the loss proportional to its own intensity, Dipi, plus the loss proportional to the intensity of the other laser, hjpj: (A.9a)n˙1=J1qd1-Γ1vg1a1(n1-n01)p1-n1τe1,(A.9b)n˙2=J2qd2-Γ2vg2a2(n2-n02)p2-n2τe2,(A.9c)p˙1=(Γ1vg1a1(n1-n01)-1τp1-D1p1-h2p2)p1,(A.9d)p˙2=(Γ2vg2a2(n2-n02)-1τp2-D2p2-h1p1)p2,where Di (i=1,2) is the second order loss constant of the isolated ith laser and hi is the coupling strength measuring the cross-loss effect of ith laser on jth laser.
Thus according to (A.9a)–(A.9d), two lasers happen to be cross-coupled through their resonators, so that each of them can modulate the cavity loss of the other. Technically, intensity-dependent intrinsic and cross-losses may be implemented, for example, using an intracavity electro-optic modulator fed by a current proportional to the output power [37, 38].
We will consider the pump currents J1 and J2, and the coupling strengths h1 and h2, as free parameters of the model. For the present, we cannot judge with any confidence the numerical value of D1,2, however, as it is demonstrated in the main body of the paper, the exact value of the intrinsic second order loss is not all that critical and does not affect principal results of our analysis, providing that this parameter is small in a sense. For the purposes of model calculations, we adopt D1,2 to be somewhat less than 4×10-5cm3/s.
Presumably, Hofelich-Abate and Hofelich [39] were first to introduce (in general form) an intensity-dependent self-limitation term in the photon-density rate equation. Later, that has been done in an explicit form of the second order loss [40, 41]. Those and subsequent studies [42, 43] showed efficient damping of relaxation oscillations in the presence of intensity-dependent losses.
As to the formal analysis of laser coupling via intensity-loss cross-modulation, very few attempts have been done so far to consider such a mechanism—for example, [44, 45]. The former work, however, does not treat intrinsic second order losses.
The works which address loss-coupled modes of a single laser rather than loss-coupled single-mode lasers are much more plentiful and varied, and we refer the reader for the reviews in [46, ch. 8] and [47, ch. 12]. In his seminal paper [18], Baer studied multimode regime of Nd:YAG (neodymium-doped yttrium aluminum garnet) laser with the intracavity-doubling KTP (potassium titynal phosphate) crystal both experimentally and numerically, and proposed the following rate equations for two coupled longitudinal modes: (A.10a)τfG˙1=G10-(βI1+β12I2+1)G1,(A.10b)τfG˙2=G20-(βI2+β21I1+1)G2,(A.10c)τeI˙1=(G1-α1-εI1-2εI2)I1,(A.10d)τeI˙2=(G2-α2-εI2-2εI1)I2,where Gi and Ii (i=1,2) are the respective gain and intensity of ith mode, τf is the fluorescence lifetime, Gi0 is the small-signal gain (pump parameter) for ith mode, β is the saturation parameter which determines how strongly the intensity depletes the available gain, β12=β21 is the cross-saturation parameter for modes 1 and 2, τe is the cavity round-trip time, αi is the loss of ith mode, and ε is the nonlinear coupling coefficient, which models the intracavity-doubling crystal as an intensity-dependent loss in the laser resonator.
The last two terms in (A.10c) and (A.10d) represent second order losses that are due to intracavity second-harmonic generation and sum-frequency generation, respectively. Numerical calculations revealed antiphase synchronous oscillations in (A.10a)–(A.10d). In antiphase state, either mode has precisely the same time profile being shifted by 1/2 of a period from its counterpart. This type of dynamics was later observed in a multimode Nd3+:YAG laser with intracavity doubling crystal [48]. For n antiphase oscillators, the phase shift between any nearest neighbors is 1/n of a period.
At β12=β21=0, system (A.10a)–(A.10d) is essentially identical to (A.9a)–(A.9d), except that intensity-dependent intrinsic and cross-losses in (A.9c) and (A.9d) are allowed to be independent.
The scaling
(A.11)u1=Γ1vg1a1τp1(n1-n01)-1,u2=Γ2vg2a2τp2(n2-n02)-1,v1=Γ1vg1a1τe1p1,v2=Γ2vg2a2τe2p2,γ1=Γ1vg1a1τp1(J1τe1qd1-n01)-1,γ2=Γ2vg2a2τp2(J2τe2qd2-n02)-1,δ1=D1τp1Γ1vg1a1τe1,δ2=D2τp2Γ2vg2a2τe2,ϰ1=h1τp2Γ1vg1a1τe1,ϰ2=h2τp1Γ2vg2a2τe2,β=τe2τe1,ε1=τp1τe1,ε2=τp2τe2,t′=tτe1
turns (A.9a)–(A.9d) into nondimensional form (A.7a)–(A.7d). Since the lasers are made of the same material, we can put d1=d2, Γ1=Γ2, vg1=vg2, a1=a2, n01=n02, τe1=τe2, τp1=τp2, and D1=D2, whence β=1, ε1=ε2=ε, and δ1=δ2=δ, and we arrive, dropping the prime at t, at the set of (1a)–(1d).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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