We study a general discrete planar system for modeling stage-structured
populations. Our results include conditions for the global convergence of
orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are
periodic we obtain weaker conditions for extinction. We also study a rational
special case of the system for Beverton-Holt type interactions and show that
the persistence equilibrium (in the positive quadrant) may be globally
attracting even in the presence of interstage competition. However, we
determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and the system exhibits period two
oscillations.
1. Introduction
Stage-structured models of single-species populations with lowest dimension in discrete time are expressed as planar systems of difference equations. For a general expression of these models, consider system(1a)At+1=s1tσ1c11tJt,c12tAtJt+s2tσ2c21tJt,c22tAtAt,(1b)Jt+1=btϕc1tJt,c2tAtAtfrom [1] in which J(t) and A(t) are numbers (or densities) of juveniles and adults, respectively, remaining after t (juvenile) periods. The vital rates si and b (survival and inherent low density fertility) as well as the competition coefficients ci and cij in (1a) and (1b) may be density dependent; that is, they may depend on J and A and also explicitly on time; that is, the system may be nonautonomous. Early examples of matrix models used in species populations dynamics can be found in [2–5] and their comprehensive treatment is provided in [6].
Under certain constraints on the various functions, including periodic vital rates and competition coefficients having the same common period p, sufficient conditions for global convergence to zero (extinction) as well as the existence of periodic orbits for (1a) and (1b) are established in [1]. If μ is the mean fertility rate (the mean value of b(t) above), then it is also shown that orbits of period p appear when μ exceeds a critical value μc, while global convergence to 0 or extinction occurs when μ<μc. On the other hand, conditions under which the species survives (i.e. permanence) were studied in [7, 8].
In this paper, we study the following abstraction of the matrix model (1a) and (1b): (2a)xn+1=σ1,nxn,ynyn+σ2,nxn,ynxn,(2b)yn+1=ϕnxn,ynxn,where for each time period n≥0 the functions σ1,n,σ2,n,ϕn:[0,∞)2→[0,∞) are bounded on the compact sets in [0,∞)2. This feature allows for (0,0) to be a fixed point of the system and it is true if, for example, σ1,n, σ2,n, ϕn are continuous functions for every n. Under biological constraints on the parameters, we may think of xn=A(n) and yn=J(n) as in (1a) and (1b).
System (2a) and (2b) includes typical stage-structured models in the literature. For instance, the tadpole-adult model for the green tree frog Hyla cinerea population that is proposed in [9] may be expressed as(3a)xn=yna+k1yn+xnc+k2xn,(3b)yn=bnxn.
This is a system of type (2a) and (2b) with Beverton-Holt type functions σ1 and σ2. Competition in (3a) and (3b) occurs separately among juveniles and adults but not between the two classes, as they feed on separate resources; thus σ1 and σ2 do not depend on both juvenile and adult numbers and ϕ is independent of both numbers. Two cases are analyzed in [9]: (i) continuous breeding with constant bn=b so that (3a) and (3b) is autonomous and (ii) seasonal breeding where bn is periodic. In addition to considering extinction and survival in the autonomous case, it is shown that seasonal breeding may be deleterious (relative to continuous breeding) for populations with high birth rates, but it can be beneficial with low birth rates.
Another system of type (2a) and (2b) is the autonomous stage-structured model with harvesting that is discussed in [10, 11], which may be written as(4a)xn+1=1-hjsjyn+1-hasaxn,(4b)yn+1=xnf1-haxn.
The numbers hj,ha∈[0,1] denote the harvest rates of juveniles and adults, respectively. The stock-recruitment function f:[0,∞)→[0,∞) may be compensatory (e.g., Beverton-Holt [12]) or overcompensatory (e.g., Ricker [13]). Compensatory recruitment is used in populations where recruitment increases with increase in densities before reaching an asymptote, while in overcompensatory models recruitment declines as density increases (see [11, 14]). A thorough analysis of the dynamics of (4a) and (4b) with the Ricker function appears in [10]. The results in [10, 11] clarify many issues with regard to the effects of harvesting in stage-structured models such as global convergence to 0 and the existence of a stable survival equilibrium as well as the so-called hydra effect for different harvesting scenarios and with different recruitment functions; this refers to the counter-intuitive situation where an increase in the harvest or mortality rate results in a corresponding increase in the total population; for example, see [15–17].
Also studied in [10] is the occurrence of periodic and nonperiodic attractors and chaotic behavior for certain parameter ranges.
Next, the model in [18] studies the harvesting and predation of sex- and age-structured populations. Although the added stage for two sexes results in a three-dimensional model, the existence of an attracting, invariant planar manifold reduces the study of the asymptotics of the system to that of the planar system:(5a)xn+1=psYyn+sxn,(5b)yn+1=xnfyn+xnp,where the density-dependent per capita reproductive rate f may be Beverton-Holt or Ricker similarly to f in (4b). Here xn is the number of females and yn is the number of young members in the population (the male population is a fixed proportion of the females).
We also mention the adult-juvenile model(6a)xn+1=s1yn,(6b)yn+1=xnfxn,ynin which all adults are removed through harvesting, predation, migration, or just dying after one period, as in the case of semelparous species, that is, an organism that reproduces only once before death. In [19] conditions for the global attractivity of the positive fixed point and the occurrence of two cycles for (6a) and (6b) are obtained. A significant difference between (5a), (5b), (6a), and (6b) and systems (3a), (3b), (4a), and (4b) is the fact that yn+1 in (5b) or in (6b) may depend on both xn and yn.
We study the qualitative properties of the orbits of (2a) and (2b) such as uniform boundedness and global convergence to 0 under minimal restrictions on time-dependent parameters. Biological constraints may be readily imposed to obtain special cases relevant to population models.
We also investigate convergence to zero with periodic parameters (extinction in a periodic environment). In particular, we show that convergence to zero occurs even if the mean value of σ2,n exceeds 1, a situation that cannot occur if σ2,n is constant in n; see Remark 16 below.
In the final section we study the dynamics of a rational special case of (2a) and (2b). Sufficient conditions for the global asymptotic stability of a fixed point in the positive quadrant [0,∞)2 as well as conditions for the occurrence of orbits of prime period two are obtained. In particular, we establish that a sufficiently high level of interspecies competition tends to destabilize the survival fixed point and result in periodic oscillations.
Discrete population models generally have been studied by numerous authors; see, for example, [20–32] and the references therein.
2. Uniform Boundedness of Orbits
Conditions under which the orbits of (2a) and (2b) are bounded are not transparent. In this section we obtain general results about the uniform boundedness of orbits of (2a) and (2b) in the positive quadrant 0,∞2. We begin with a simple, yet useful lemma.
Lemma 1.
Let α>0, let 0<β<1, and let x0≥0. If for all n≥0(7)xn+1≤α+βxnthen for every ɛ>0 and all sufficiently large values of n(8)xn≤α1-β+ɛ.
Proof.
Let u0=x0 and note that every solution of the linear, first-order equation un+1=α+βun converges to its fixed point α/(1-β). Furthermore, (9)x1≤α+βx0=α+βu0=u1,x2≤α+βx1≤α+βu1=u2and, by induction, xn≤un. Since un→α/(1-β) for every ɛ>0 and all sufficiently large n,(10)xn≤un≤α1-β+ɛ.
Theorem 2.
Let σ1,n, σ2,n, ϕn be bounded on the compact sets in [0,∞)2 for each n=0,1,2,… and suppose that for some r,M>0(11)supu,v∈0,r2σ2,nu,v≤M∀n≥0;that is, the sequence of functions {σ2,n} is uniformly bounded on the square [0,r]2. If there are numbers M0,M1>0 and σ¯∈(0,1) such that uniformly for all n(12)uϕnu,v≤M0ifu,v∈0,∞2,(13)σ1,nu,v≤M1ifu,v∈0,∞×0,M0,(14)σ2,nu,v≤σ¯ifu,v∈r,∞×0,M0,then all orbits of (2a) and (2b) are uniformly bounded and for all sufficiently large values of n satisfy(15)0≤xn≤M0M1+rM+σ¯1-σ¯,yn≤M0.
Proof.
By (2b) and (12) yn≤M0 for n≥1 so by (2a) and (13)(16)0≤xn+1≤M0M1+σ2,nu,vxn.
By (11) and (14)(17)0≤xn+1≤M0M1+maxσ¯xn,Mr≤σ¯xn+M0M1+rM.
Next, applying Lemma 1 with ɛ=σ¯/(1-σ¯), we obtain for all (large) n(18)0≤xn≤M0M1+rM1-σ¯+ɛ=M0M1+rM+σ¯1-σ¯as stated.
Corollary 3.
For functions σ1,n, σ2,n, ϕn defined on [0,∞)2 for n=0,1,2,… assume that there are numbers M0,M1>0 and σ¯∈(0,1) such that for all (u,v)∈[0,∞)2 and all n(19)uϕnu,v≤M0,σ1,nu,v≤M1,σ2,nu,v≤σ¯.Then all orbits of (2a) and (2b) are uniformly bounded and for all sufficiently large values of n(20)0≤xn≤M0M1+σ¯1-σ¯,yn≤M0.
Theorem 2 is more general than the preceding corollary. For instance, Corollary 3 does not apply to system (21)xn+1=axn+byn21+cxn,yn+1=αxn1+βxn+γyn.
However, if a∈(0,1), b,α,β>0, and c,γ≥0, then all orbits of this system with initial values in [0,∞)2 are uniformly bounded by Theorem 2.
3. Global Attractivity of the Origin
In this section we obtain general sufficient conditions for the convergence of all orbits of the system to (0,0). For population models these yield conditions that imply the extinction of species.
3.1. General Results
We start with the following lemma; see [33] for the proof and some background on this result.
Lemma 4.
Let α∈(0,1) and assume that the functions fn:0,∞k+1→0,∞ satisfy the inequality(22)fnu0,…,uk≤αmaxu0,…,ukfor all (u0,…,uk)∈[0,∞) and all n≥0. Then for every solution {xn} of the difference equation(23)xn+1=fnxn,xn-1,…,xn-kthe following is true:(24)xn≤αn/k+1maxx0,x-1,…,x-k.
Note that (22) implies that xn=0 is a constant solution of (23) and furthermore (24) implies that this solution is globally exponentially stable.
Throughout this section we assume that σi,n, ϕn are all bounded functions for i=1,2 and every n=0,1,2,…. Then the following are well-defined sequences of real numbers:(25)σ¯i,n=supu,v≥0σi,nu,v,ϕ¯n=supu,v≥0ϕnu,v.
Theorem 5.
If the inequality(26)limsupn→∞σ¯1,nϕ¯n-1+σ¯2,n<1holds, then limn→∞xn=0 for every orbit {(xn,yn)} of the planar system (2a) and (2b) in the positive quadrant 0,∞2. If also either the sequence {ϕ¯n} is bounded or the inequality(27)liminfn→∞σ¯1,n>0,holds, then every orbit of (2a) and (2b) converges to (0,0).
Proof.
By (26) there is δ∈(0,1) such that σ¯1,nϕ¯n-1+σ¯2,n≤δ for all (large) n. From (2a)(28)yn≤ϕ¯n-1xn-1so for all (large) n (2b) yields(29)xn+1≤ϕ¯n-1σ¯1,nxn-1+σ¯2,nxn≤σ¯1,nϕ¯n-1+σ¯2,nmaxxn,xn-1≤δmaxxn,xn-1.
Lemma 4 now implies that limn→∞xn=0. Furthermore either by hypothesis there is a positive number μ such that ϕ¯n≤μ or by (27) there is a positive number ρ such that σ¯1,n≥ρ for all (large) n so that(30)ϕ¯n-1≤δ-σ¯2,nσ¯1,n≤δρfor all sufficiently large values of n. Now, if M=μ or M=δ/ρ as the case may be, then from (2b) in the planar system we see that(31)limn→∞yn≤limn→∞ϕ¯n-1xn-1≤Mlimn→∞xn-1=0and the proof is complete.
Remark 6.
(1) Theorem 5 is valid even if the separate sequences {σ1,n} or {ϕ¯n} are not bounded by 1 as long as for all n large enough σ¯1,nϕ¯n-1≤δ-σ¯2,n.
(2) If (26) is satisfied but {ϕ¯n} is unbounded and {σ¯1,n} does not satisfy (27) then yn may not converge to 0; see the example following Corollary 18 below.
We consider an application of Theorem 5 to “noisy” autonomous system next. Let ɛn, ɛi,n, i=1,2, be bounded sequences of real numbers and let(32)ɛ¯=supn≥1ɛn,ɛ¯i=supn≥1ɛi,n,i=1,2.
Additionally, let σ1,σ2,ϕ:[0,∞)2→[0,∞) be bounded functions and denote their supremums over [0,∞)2 by σ¯1, σ¯2, ϕ¯, respectively. If in (2a) and (2b) we have(33)ϕnxn,yn=ϕxn,yn+ɛn,σi,nxn,yn=σixn,yn+ɛi,n,i=1,2then we refer to (2a) and (2b) as an autonomous system with low-amplitude disturbances or fluctuations in the rates σ1, σ2, ϕ, assuming that all three of these are positive functions and for all u,v≥0(34)ɛ¯≤ϕu,v,ɛ¯i≤σiu,v,i=1,2.
These inequalities ensure that the functions ϕn and σi,n are positive, as required for (2a) and (2b).
Corollary 7.
Suppose that (2a) and (2b) is an autonomous system with low-amplitude disturbances or fluctuations in the above sense. If(35)σ¯1+ɛ¯1ϕ¯+ɛ¯+σ¯2+ɛ¯2<1then the origin is the unique, globally asymptotically stable fixed point of (2a) and (2b) relative to the positive quadrant [0,∞).
Note that (35) holds for nontrivial sequences ɛn, ɛi,n of real numbers if σ¯1ϕ¯+σ¯2<1.
Remark 8.
Since in the above discussion the sequences ϵn, ϵi,n, i=1,2, are arbitrary bounded sequences, they can also be sequences of random variables that are drawn from distributions with finite support. For example, ϵn, ϵi,n can be drawn from uniform distribution on some interval [0,θ] so long as(36)σ¯1+θϕ¯+θ+σ¯2+θ<1.
Corollary 7 will hold, implying that the origin is globally attracting even in the presence of noise.
In the autonomous case where the three parameter functions σ1,n, σ2,n, ϕn do not depend on n at all, we have the following planar system: (37a)xn+1=σ1xn,ynyn+σ2xn,ynxn,(37b)yn+1=ϕxn,ynxn.
If in Corollary 7 we set ɛ¯i,ɛ¯=0 in (35) then we obtain the following result for the above autonomous system.
Corollary 9.
Assume that σ1,σ2,ϕ:[0,∞)2→[0,∞) are bounded functions and the following inequality holds:(38)σ¯1ϕ¯+σ¯2<1;then the origin is the unique, globally asymptotically stable fixed point of (37a) and (37b) relative to the positive quadrant [0,∞)2.
Inequality (38) may be explicitly related to the local asymptotic stability of the origin for (37a) and (37b) when the functions σ1, σ2, ϕ are smooth. Consider the associated mapping(39)Fu,v=uσu,v+vσ1u,v,uϕu,vwhose linearization at (0,0) has eigenvalues(40)λ±=σ20,0±σ20,02+4σ10,0ϕ0,02.
These are real and a routine calculation shows that λ±<1 if(41)σ10,0ϕ0,0+σ20,0<1.
Under suitable differentiability hypotheses, this inequality is implied by (38) and is equivalent to it if the suprema of σ2 and σ1ϕ occur at (0,0).
3.2. Folding the System
In the next and later sections it will be convenient to fold system (2a) and (2b) to a second-order equation; see [34] for more details on folding. System (2a) and (2b) in general folds as follows: substitute for yn+1 from (2b) into (2a) to obtain(42)xn+2=σ1,n+1xn+1,ϕnxn,hnxn,xn+1xnϕnxn,hnxn,xn+1xn+σ2,n+1xn+1,ϕnxn,hnxn,xn+1xnxn+1,where(43)hnxn,xn+1=ynis derived by solving (2a) for yn. Although an explicit formula for hn is not feasible in general, it is readily obtained in typical cases; for instance, suppose that σ2,n(u,v)=σ2,n(u) and σ1,n(u,v)=σ1,n(u) are both independent of (or constant in) v for all n; note that systems (3a), (3b), (4a), (4b), (5a), (5b), (6a), and (6b) are all of this type. In this case it is clear that(44)yn=hnxn,xn+1=xn+1-σ2,nxnxnσ1,nxnand furthermore (42) reduces to(45)xn+2=σ1,n+1xn+1ϕnxn,xn+1-σ2,nxnxnσ1,nxnxn+σ2,n+1xn+1xn+1.
The pair of first-order equations (44) and (45) represents folding of (2a) and (2b). Note that with positive parameter functions, each pair x0,y0≥0 generates an orbit {(xn,yn)} of (2a) and (2b) that is in [0,∞)2 for all n. So we have xn+1,xn≥0 and also by (43) hn(xn,xn+1)≥0 so ϕn(xn,hn(xn,xn+1)) is well defined for every such orbit of (2a) and (2b).
Remark 10.
An even simpler reduction than the above is possible if ϕn(u,v)=ϕn(u) is independent of (or constant in) v. In this case,(46)xn+2=σ1,n+1xn+1,ϕnxnxnϕnxnxn+σ2,n+1xn+1,ϕnxnxnxn+1and it is not necessary to solve (2a) for yn implicitly (i.e., the system folds without inversions). Special cases of this type include systems (3a), (3b), (4a), and (4b).
3.3. Global Convergence to Zero with Periodic Parameters
The results in this section show that global convergence to zero may occur even if (26) does not hold; see Remark 16 below. Recall from the proof of Theorem 5 that(47)xn+1≤σ1,nϕ¯n-1xn-1+σ¯2,nxn.
The right-hand side of the above inequality is a linear expression. Consider the linear difference equation(48)un+1=anun+bnun-1,an+p1=an,bn+p2=bn,where the coefficients an and bn are nonnegative and their periods p1 and p2 are positive integers with least common multiple p=lcm(p1,p2); we say that the linear difference equation (48) is periodic with period p. In this study we assume that(49)an,bn≥0,n=0,1,2,….
By Lemma 4 every solution of (48) converges to zero if an+bn<1 for all n. However, it is known that convergence to zero may occur even when an+bn exceeds 1 (for infinitely many n in the periodic case). We use the approach in [35] to examine the consequences of this issue when the planar system has periodic parameters. The following result is an immediate consequence of Theorem 13 in [35].
Lemma 11.
Assume that αj, βj for j=1,2,…,p are obtained by iteration from (48) from the real initial values:(50)α0=0,α1=1;β0=1,β1=0.
Suppose that the quadratic polynomial(51)αpr2+βp-αp+1r-βp+1=0is proper, that is, not 0=0, and suppose that it has a real root r1≠0. If the recurrence(52)rn+1=an+bnrngenerates nonzero real numbers r2,…,rp then {rn}n=1∞ is periodic with preiod p and yields a triangular system of first-order equations that is equivalent to (48) as follows:(53)tn+1=-bnrntn,t1=u1-r1u0,(54)un+1=rn+1un+tn+1.
System (53) and (54) is also known as a semiconjugate factorization of (48); see [36] for an introduction to this concept. The sequence {rn} that is generated by (52) is said to be (unitary) eigensequence of (48). Eigenvalues are essentially constant eigensequences for if p=1 in Lemma 11 then (51) reduces to (55)α1r2+β1-α2r-β2=0,r2-a1r-b1=0and the latter equation is the standard characteristic equation of (48) with constant coefficients; see [35] for more details on the semiconjugate factorization of linear difference equations.
Each of (53) and (54) readily yields a solution by iteration as follows:(56)tn=t1-1n-1b1b2⋯bn-1r1r2⋯rn-1,(57)un=rnrn-1⋯r2u1+rnrn-1⋯r3t2+⋯rntn-1+tn=rnrn-1⋯r2r1u0+∑i=1n-1rnrn-1⋯ri+1ti+tn.
Lemma 12.
Suppose that the numbers αn and βn are defined as in Lemma 11, though we do not assume that (48) is periodic here. Then
βn=0 for all n≥2 if and only if b1=0;
if (49) holds then for all n≥2(58)αn≥a1a2⋯an-1,βn≥b1a2⋯an-1,(59)α2n-1≥b2b4⋯b2n-2,β2n≥b1b3⋯b2n-1.
Proof.
(a) Let b1=0. Then β2=b1=0 and since β1=0 by definition it follows that β3=0. Induction completes the proof that βn=0 if n≥2. The converse is obvious since b1=β2.
(b) Since α2=a1 and β2=b1 the stated inequalities hold for n=2. If (58) is true for some k≥2 then (60)αk+1=akαk+bkαk-1≥akαk≥a1a2⋯ak-1ak,βk+1=akβk+bkβk-1≥akβk≥b1a2⋯ak-1ak.
Now, the proof is completed by induction. The proof of (59) is similar since(61)α3=a2α2+b2α1≥b2,β4=a3β3+b3β2≥b3b1and if (59) holds for some k≥2 then (62)α2k+1≥b2kα2k-1≥b2b4⋯b2k-2b2k,β2k+2≥b2k+1β2k≥b1b3⋯b2k-1b2k+1which establishes the induction step.
Lemma 13.
Assume that (49) holds with ai>0 for i=1,…,p and (48) is periodic with period p≥2. Then one has the following.
(a) Equation (48) has a positive (hence unitary) eigensequence {rn} of period p.
(b) If bi>0 for i=1,…,p then(63)r1r2⋯rp=12αp+1+βp+αp+1-βp2+4αpβp+1.
Hence, r1r2⋯rp<1 if(64)αpβp+1<1-αp+11-βp.
(c) If bi<1 for i=1,…,p then r1r2⋯rp>b1b2⋯bp.
Proof.
(a) Lemma 12 shows that αi>0 for i=2,…,p+1. Now, either (i) b1>0 or (ii) b1=0. In case (i), the root r+ of the quadratic polynomial (51) is positive since by Lemma 12βp+1>0 and thus(65)r+=αp+1-βp+αp+1-βp2+4αpβp+12αp>αp+1-βp+αp+1-βp2αp≥0.
If r1=r+ then from (52) ri=ai-1+bi-1/ri-1≥ai-1>0 for i=2,…,p+1. Thus, by Lemma 11, (48) has a unitary (in fact positive) eigensequence of period p. If b1=0 then by Lemma 12βp=βp+1=0 and (51) reduces to(66)αpr2-αp+1r=0which has a root r+=αp+1/αp>0. As in the previous case it follows that (48) has a positive eigensequence of period p.
(b) To establish (63), let r1=r+ and note that (51) can be written as(67)r1=αp+1r1+βp+1αpr1+βp.
Since {rn} has period p, rp+1=r1 so from (52) and the definition of the numbers αn and βn it follows that (68)ap+bprp=rp+1=αp+1r1+βp+1αpr1+βp=apαp+bpαp-1r1+apβp+bpβp-1αpr1+βp=apαpr1+βp+bpαp-1r1+βp-1αpr1+βp=ap+bpαpr1+βp/αp-1r1+βp-1.
Since bp≠0 it follows that(69)rp=αpr1+βpαp-1r1+βp-1.
We claim that if bi≠0 for i=1,…,p then(70)rp-j=αp-jr1+βp-jαp-j-1r1+βp-j-1,j=0,1,…,p-2.
This claim is easily seen to be true by induction; we showed that it is true for j=0 and if (70) holds for some j then by (52) (71)ap-j-1+bp-j-1rp-j-1=rp-j=ap-j-1αp-j-1+bp-j-1αp-j-2r1+ap-j-1βp-j-1+bp-j-1βp-j-2αp-j-1r1+βp-j-1=ap-j-1αp-j-1r1+βp-j-1+bp-j-1αp-j-2r1+βp-j-2αp-j-1r1+βp-j-1=ap-j-1+bp-j-1αp-j-2r1+βp-j-2αp-j-1r1+βp-j-1from which it follows that(72)rp-j-1=αp-j-1r1+βp-j-1αp-j-2r1+βp-j-2and the induction argument is complete. Now, using (70), we obtain(73)rprp-1⋯r2r1=αpr1+βpαp-1r1+βp-1αp-1r1+βp-1αp-2r1+βp-2⋯α2r1+β2α1r1+β1r1=αpr1+βp.
Given that r1=r+ (73) implies that (74)r1r2⋯rp=αpαp+1-βp+αp+1-βp2+4αpβp+12αp+βp=12αp+1+βp+αp+1-βp2+4αpβp+1and (63) is obtained. Hence, r1r2⋯rp<1 if(75)αp+1+βp+αp+1-βp2+4αpβp+1<2.
Upon rearranging terms and squaring,(76)αp+1-βp2+4αpβp+1<4-4αp+1+βp+αp+1+βp2which reduces to (64) after straightforward algebraic manipulations.
(c) First, assume that p is odd. Then by (59)(77)αpβp+1=b2b4⋯bp-1b1b3⋯bp=b1b2⋯bpso from (63)(78)r1r2⋯rp>αpβp+1=b1b2⋯bp.
If bi<1 for i=1,…,p then b1b2⋯bp<1 so b1b2⋯bp>b1b2⋯bp as required. Now let p be even. Then from (63) and (59)(79)r1r2⋯rp>αp+1+βp2≥b2b4⋯bp+b1b3⋯bp-12.
If bi<1 for i=1,…,p then b2b4⋯bp≥b1b2⋯bp and b1b3⋯bp-1≥b1b2⋯bp and the proof is complete.
Some of the numbers ai may exceed 1 in Lemma 13 without affecting the conclusions of the lemma. Additionally, not all the conditions in Lemma 13 are necessary. For instance, if b1=0 then Lemma 13(c) holds trivially. Additionally, by Lemma 12(a), βn=0 for n≥2 so the following equality must hold instead of (63):(80)r1r2⋯rp=αp+1.
This is in fact true because r1=r+=αp+1/αp so repeating the argument in the proof of Lemma 13(b) yields rp-j=αp-j/αp-j-1 for j=0,1,…,p-2. Hence(81)rprp-1⋯r2r1=αpαp-1αp-1αp-2⋯α2α1αp+1αp=αp+1as claimed. These observations establish the following version of Lemma 13.
Lemma 14.
Let ai>0 and let bi≥0 for i=1,…,p with b1=0. Then the linear equation (48) has a positive (hence unitary) eigensequence {rn} of period p given by(82)r1=αp+1αp,rj=αjαj-1,j=2,…,pand 0=b1b2⋯bp<r1r2⋯rp<1 if αp+1<1.
In Lemma 14 some of the numbers ai or bi may exceed 1 without affecting the conclusions of the lemma.
Theorem 15.
Assume that (27) holds and the sequences and {σ¯1,nϕ¯n-1} and {σ¯2,n} have period p with σ¯2,i>0 and σ¯1,iϕ¯i-1≥0 for i=1,…,p. Additionally let the numbers αn,βn be as previously defined with an=σ¯2,n and bn=σ¯1,nϕ¯n-1. All nonnegative orbits of the planar system converge to (0,0) if either one of the following holds:
0<σ¯1,iϕ¯i-1<1 and (64) holds.
σ¯1,1ϕ¯0=0 and αp+1<1.
Proof.
Let {un} be a solution of the linear equation (48) with an=σ¯2,n, bn=σ¯1,nϕ¯n-1, u0=x0, and u1=x1. Then by (47) (83)x2≤σ¯1,1ϕ¯0x0+σ¯2,1x1=σ¯1,1ϕ¯0u0+σ¯2,1u1=u2,x3≤σ¯1,2ϕ¯2x2+σ¯2,2x2≤σ¯1,2ϕ¯1u1+σ¯2,2u2=u3.
By induction it follows that xn≤un. If (64) holds then, by Lemma 13, limn→∞un=0 so {xn} converges to 0. Furthermore, limn→∞yn=0 as in the proof of Theorem 5 and the proof is complete.
Remark 16.
(1) Condition (64) involves the numbers αj, βj rather than the coefficients of (48) directly. In the case of period p=2 the role of ai and bi is more apparent. Inequality (64) in this case is (84)α2β3<1-α31-β2,a1a2b1<1-b2-a1a21-b1and simple manipulations reduce the last inequality to(85)a1a2<1-b11-b2.
(2) Inequality (85) holds even if a1>1 or a2>1 thus showing how global convergence to (0,0) my occur when (26) does not hold. Furthermore, it is possible that (85) holds together with(86)a1+a22>1.
Note that (85) holds even with arbitrarily large mean value in (86) if say a1→0 as a2→∞. In population models this implies that if (85) holds with an=σ¯2,n and bn=σ¯1,nϕ¯n-1 then extinction may still occur after restocking the adult population to raise the mean value of the composite parameter σ¯2,n above 1 by a wide margin.
(3) In Theorem 15 the individual sequences σ¯1,n, ϕ¯n need not be periodic or even bounded. Therefore, the theorem applies to (2a) and (2b) even if the system itself is not periodic as long as the combination σ¯1,nϕ¯n-1 of parameters is periodic along with σ¯2,n.
4. Dynamics of a Beverton-Holt Type Rational System
In this section we apply some of the preceding results and obtain some new ones to study boundedness, extinction, and modes of survival in some rational special cases of (2a) and (2b). In population models these types of systems include the Beverton-Holt type interactions. Specifically, we consider the following nonautonomous system and some of its special cases:(87a)xn+1=α1,nyn1+β1,nxn+γ1,nyn+α2,nxn1+β2,nxn+γ2,nyn,(87b)yn+1=bnxn1+c1,nxn+c2,nyn,where we assume that for all n≥0 and i=1,2(88)α1,n>0,bn,α2,n,βi,n,γi,n,ci,n≥0bn>0 for infinitely many n.
For example, if we think of αi as the natural survival rates then the population model (3a) and (3b) is a special case of (87a) and (87b). If we allow αi to include additional factors such as harvesting rates then (87a) and (87b) is an extension of the model in [11] (with a Beverton-Holt recruitment function) in the sense that the competition coefficients βi,n, γi,n, and ci,n may be nonzero as well as time-dependent.
4.1. Uniform Boundedness and Extinction
We now examine boundedness and global convergence to 0 (extinction) in (87a) and (87b). The next result is in part a consequence of Corollary 3.
Corollary 17.
Assume that (88) holds.
(a) Let the sequence {α1,n} be bounded and limsupn→∞α2,n<1. If there is M0>0 such that bn≤M0c1,n for all n larger than a given positive integer then all orbits of (87a) and (87b) are uniformly bounded.
(b) Let the sequence {bn} be bounded and suppose that there is M>0 such that(89)α1,n≤Mγ1,n,α2,n≤Mβ2,nfor all n larger than a given positive integer. Then all orbits of (87a) and (87b) are uniformly bounded.
Proof.
(a) By hypothesis, for all (large) n,(90)bnxn1+c1,nxn+c2,nyn≤M0c1,nxn1+c1,nxn+c2,nyn<M0.
By hypothesis, there is M1>0 and δ∈(0,1) such that for all u,v≥0 and all sufficiently large values of n(92)σ1,nu,v≤α1,n≤M1,σ2,nu,v≤α2,n≤δ.
Now an application of Corollary 3 completes the proof of (a).
(b) By (89) for all large n it follows that(93)α1,nyn1+β1,nxn+γ1,nyn≤Mγ1,nyn1+β1,nxn+γ1,nyn<Mand, likewise,(94)α2,nxn1+β2,nxn+γ2,nyn≤Mβ2,nxn1+β2,nxn+γ2,nyn<Mfor all large n. Therefore, xn≤2M. Next, if {bn} is bounded then yn≤2Mbn is also bounded and the proof is complete.
The next result follows readily from Theorem 5.
Corollary 18.
The origin (0,0) attracts every orbit of (87a) and (87b) in 0,∞2 if(95)limsupn→∞α1,nbn-1+α2,n<1and either bn is bounded or liminfn→∞α1,n>0.
The above corollary is false when (95) holds if bn is unbounded and thus α1,n has a subsequence that converges to 0.
Example 19.
Consider system (96)xn+1=α-nyn+sxn,yn+1=βαnxn1+cxn,where α>1, β>0, 0≤s<1, c≥0, σ1,n=α-n, and bn=βαn. Then (95) is satisfied, so limn→∞xn=0. But yn does not approach 0 for large enough α; this may be inferred from Lemma 4 which shows that xn converges to 0 at an exponential rate δn/2 where δ=s+β/α∈(0,1). Thus(97)yn=1α-nxn+1-sxn=αnxn+1-sxnwill not converge to 0 if α is sufficiently large.
Corollary 17 takes a simpler form for the autonomous special case of (87a) and (87b); namely,(98a)xn+1=α1yn1+β1xn+γ1yn+α2xn1+β2xn+γ2yn,(98b)yn+1=bxn1+c1xn+c2ynwith constant parameters (99)α1,b>0,α2,βi,γi,ci≥0.
The following result is applicable to (3a) and (3b) as well as special cases of (4a), (4b), (5a), and (5b) with rational f.
Corollary 20.
Assume that (99) holds. All orbits of (98a) and (98b) in 0,∞2 are uniformly bounded if either one of the following conditions holds:
α2<1 and c1>0.
γ1,β2>0.
It is noteworthy that if in part (a) above c1=0 then (98a) and (98b) may have unbounded solutions as in, for example, system (100)xn+1=α1yn,yn+1=bxn1+c2yn,where α2=c1=0 and the remaining parameters are positive. This system folds to the second-order rational equation(101)xn+2=α12bxnα1+c2xn+1which is known to have unbounded solutions if α1b>1; see [37].
Corollary 18 likewise simplifies in the autonomous case.
Corollary 21.
Assume that (99) holds with α1b+α2<1. Then the origin (0,0) is the globally asymptotically stable fixed point of (98a) and (98b) relative to 0,∞2.
4.2. Persistence and the Role of Competition
We now explore the effects of competition in the autonomous system (98a) and (98b). There are 6 different competition coefficients and to reduce the number of different cases we focus on the special case below where βi,γi=0:(102)xn+1=α1yn+α2xn,(103)yn+1=bxn1+c1xn+c2yn.
If αi define the natural survival rates si, then this system is complementary to (3a), (3b), (4a), and (4b) in the sense that in both of those systems c2=0.
By the last two corollaries, all orbits of the rational system (102) and (103) in [0,∞)2 are uniformly bounded if c1>0 and α2<1 and they converge to the origin if α1b+α2<1. We now examine this rational system in more detail using its folding, namely, the second-order rational equation(104)xn+2=axn+1+σxn1+Axn+1+Bxn,where(105)a=α2,σ=α1b,A=c2α1,B=1α1α1c1-α2c2.
See (45); y-component is given by (44) or calculated directly using (102) as(106)yn=1α1xn+1-α2xn.
With initial values x0 and x1=α1y0+α2x0 derived from (x0,y0)∈0,∞2, x-component of the orbits {(xn,yn)} of the system is obtained by iterating (104). The equation in (106) is passive in the sense that after x-component of the orbit is generated by the core equation (104) y-component is derived from (106) without any further iterations. This observation also establishes the nontrivial fact that solutions of (104) that correspond to the orbits of the system in 0,∞2 are nonnegative and well-defined even for B<0.
If α1b+α2<1, that is, σ<1-a, then zero is the only fixed point of (104). Corollary 21 establishes that, in this case, zero is globally asymptotically stable relative to [0,∞). On the other hand, when α1b+α2>1, that is, σ>1-a, then 0 is no longer a stable fixed point of (104). By routine calculations, one can show that zero is a saddle point when 1-a<σ<1+a and if σ>1+a then zero is a repeller.
In addition, when σ>1-a and a=α2<1, system (102) and (103) also has a fixed point in (0,∞)2 given by(107)x¯=σ-1-a1-aA+B=α1α1b+α2-11-α2α1c1+1-α2c2,y¯=1-α2α1x¯.
We note that x¯ is also a positive fixed point of folding (104). Under certain conditions, x¯ attracts all solutions of (104) with positive initial values, and it is thus a survival equilibrium. We state the following result from literature; see [38].
Lemma 22.
Let I be an open interval of real numbers and suppose that f∈C(Im,R) is nondecreasing in each coordinate. Let x¯∈I be a fixed point of the difference equation(108)xn+1=fxn,xn-1,…,xn-m+1and assume that the function h(t)=f(t,…,t) satisfies the conditions(109)ht>tift<x¯,ht<tift>x¯,t∈I.Then I is an invariant interval of (108) and x¯ attracts all solutions with initial values in I.
Theorem 23.
Let a<1<a+σ; that is, α2<1<α1b+α2. If the function(110)fu,v=au+σvAu+Bv+1is nondecreasing in both arguments, then the fixed point x¯ attracts all solutions of (104) with initial values in (0,∞).
Proof.
If we let(111)ht=at+σt1+A+Btthen the fixed point x¯ is the solution of h(t)=t. For t>0, we may write h(t)=ϕ(t)t where(112)ϕt=a+σ1+A+Btwith ϕx¯=hx¯x¯=1.
Now,(113)ϕ′t=-σA+B1+A+Bt2<0for all t>0, so ϕ(t) is strictly decreasing for all t>0. Therefore, (114)t<x¯implies that ht=ϕtt>ϕx¯t=t,t>x¯implies that ht=ϕtt<ϕx¯t=t.
If α1b+α2>1 and c2=0 then A=0, so fu,fv>0. Therefore by Theorem 23x¯ is globally asymptotically stable. However, if c2>0, then fu may not be positive, so the results of Theorem 23 may not apply to this case. The next result shows that orbits of the system may converge to x¯ if c2>0 but not too large.
Theorem 24.
Let c1>0 and let a<1<a+σ; that is, α2<1<α1b+α2. Then there exists c>0 such that for c2∈[0,c] the fixed point x¯ of (104) is globally asymptotically stable relative to (0,∞).
Proof.
Since(116)fu=a-AσvAu+Bv+12=aAu+Bv2+2Aau+a+2aB-AσvAu+Bv+12to ensure that fu≥0 it suffices for 2aB-Aσ≥0; that is,(117)2α2α1c1-α2c2-c2α1b≥0which is equivalent to(118)c2≤2α1α2c1α1b+2α22≐cand the proof is complete.
If c2 is sufficiently large then fu is not positive on (0,∞). Furthermore, x¯ also becomes unstable for large enough c2, which we establish next by examining the linearization of (104) around x¯.
The characteristic equation associated with the linearization of (104) at x¯ is given by(119)λ2-pλ-q=0,where(120)p=fux¯,x¯=a-1-aAx¯1+A+Bx¯,q=fvx¯,x¯=σ-1-aBx¯1+A+Bx¯.
The roots of (119) are given by(121)λ1=p-p2+4q2,λ2=p+p2+4q2.
Since fv(u,v)>0 for all u,v∈(0,∞) it follows that q>0 and both roots are real with λ1<0 and λ2>0. Furthermore, λ2<1 if(122)p+p2+4q2<1that is q<1-pwhich is equivalent to(123)21-aA+Bx¯>σ-1-a.
This inequality holds, since x¯>0 under our assumptions on the parameters. Therefore, λ2<1. On the other hand, λ1>-1 if and only if(124)p-p2+4q2>-1that is p+1>qwhich is equivalent to(125)2Aa+Bx¯>σ-1+a.
Note that when (1-a)<σ<(1+a) this is trivially the case since x¯>0 under our assumptions on the parameters. Thus, x¯ is locally asymptotically stable if σ<1+a.
Next, λ1<-1 if σ>1+a and(126)2Aa+Bx¯<σ-1+a.
We summarize the above results in the following lemma.
Lemma 25.
Let a<1<a+σ; that is, α2<1<α1b+α2. Then the fixed point x¯ of (104) is
locally asymptotically stable if and only if (125) holds. In particular, this is true if(127)1-a<σ<1+a,thatis1-α2<α1b<1+α2.
It is a saddle point if and only if (126) holds with σ>1+a; that is α1b>1+α2.
Inequality (126) implies a range for c2 that we now determine. Let(128)k=σ-1+aσ-1-a<1.
Then k∈(0,1) if σ>1+a,(129)2Aa+Bx¯<σ-1+a⟹2Aa+BA+B<σ-1+aσ-1-a1-a=1-ak.
Since (130)2Aa+B=2α1c2α2+c1α1-c2α2=2c1,A+B=1α1c1α1+1-α2c2
(129) is equivalent to(131)2c1α1c1α1+1-α2c2<1-ak=1-α2k.
From the above inequality we obtain(132)c2>α1c12-1-α2k1-α22k≐c¯.
Thus if c2>c¯ then x¯ is a saddle point and in particular the fixed point (x¯,y¯) is unstable. These observations lead to the following which may be compared with Theorem 24.
Corollary 26.
Assume that (99) holds for system (102) and (103) and α2<1<α1b+α2. Then the fixed point (x¯,y¯) is unstable if c2>c¯.
Our final result establishes that when c2>0 is sufficiently large system (102) and (103) can have a prime period two orbit which occurs as x¯ becomes unstable. Existence of periodic orbits is established via the folding in (104).
The difference equation in (104) has a positive prime period two solution if there exist real numbers m,M>0 and m≠M such that(133)m=fM,m,M=fm,M;that is,(134)m=aM+σmAM+Bm+1,M=am+σMAm+BM+1from which we get(135)m-aMAM+Bm+1=σm,M-amAm+BM+1=σM;that is,(136)AmM+Bm2+m-AaM2-aBMm-aM=σm,(137)AmM+BM2+M-Aam2-aBMm-am=σM.
Taking the difference of the right and left sides of (136) and (137) yields (138)Bm2-M2+m-M-AaM2-m2-M-m=σm-M,B+Aam-Mm+M=σ-1+am-M.
When m≠M, we get(139)B+Aam+M=σ-1+aand since the left side of the last equation is positive this implies that σ-(1+a)>0. Stated differently, if σ-(1+a)<0, then (104) cannot have a positive prime period two solution.
Similarly, taking the sum of the right and left sides of (136) and (137) yields(140)2AmM+Bm2+M2+m+M-Aam2+M2-2aBMm-am+M=σm+M.
Adding and subtracting 2(B-Aa) to and from the left hand side of the last expression in (140) yields(141)2A-aB-B+AaMm+B-Aam+M2=σ-1-am+M;that is,(142)21+aA-BMm=σ-1-am+M-B-Aam+M2=m+Mσ-1-a-B-Aam+M=m+Mσ-1-a-B-Aaσ-1+aB+Aa=m+MAa+BB+Aaσ-1-a-B-Aaσ-1+a.
Simplifying the right hand side, it follows that(143)1+aA-BMm=σ-1+aAa+B2Aaσ-1+aB.
Now, since we are assuming that σ-(1+a)>0, then σ-1>0, so the right side of (143) is positive, which implies that A-B>0. Stated differently, if A<B, then (104) has no positive prime period two solution.
From (143) we get(144)Mm=σ-1+aAaσ-1+aBc1+aA-BAa+B2≔Qand let m+M=P, from which we obtain that M=P-m and m=P-M. This means that(145)mP-m=Q,MP-M=Q;that is, m and M are the roots of the quadratic(146)St=t2-Pt+Q,where P,Q>0 and(147)t±=P±P2-4Q2.
To ensure that m and M are real, the roots of S(t) must be real, which is the case if and only if P2-4Q>0; that is,(148)σ-1+aσ-1+a-4Aaσ-1+aB1+aA-B>0.
We summarize the above results as follows.
Theorem 27.
The second-order difference equation in (104) has a positive prime period two solution if and only if all of the following conditions are satisfied:
σ-(1+a)>0.
A-B>0.
(σ-(1+a))σ-(1+a)-4(Aa(σ-1)+aB)/(1+a)(A-B)>0.
The next result shows that a solution of period two appears when x¯ loses its stability.
Corollary 28.
The second-order difference equation in (104) has a positive prime period two solution if and only if x¯ is a saddle point.
Proof.
Suppose x¯ is a saddle point. Then, by Lemma 25(b), 2(Aa+B)x¯<σ-(1+a) from which we infer that σ-(1+a)>0.
Now 2(Aa+B)x¯<σ-(1+a) implies that(149)2Aa+B1-aA+Bσ-1-a<σ-1+awhich is true if and only if(150)2Aa+Bσ-1-a<1-aA+Bσ-1+a.
Adding and subtracting (1+a)(A-B)(σ-(1+a)) to and from the right hand side of the last expression in (150) yields (151)1+aA-Bσ-1+a+σ-1+a1-aA+B-1+aA-B=1+aA-Bσ-1+a+σ-1+a2B-2Aa.
Therefore, (152)2Aa+Bσ-1-a+2Aa-Bσ-1+a=4Aaσ-1+aB<1+aA-Bσ-1+a;that is,(153)1+aA-Bσ-1+a-4Aaσ-1+aB>0from which we infer that A-B>0 and the roots of S(t) are guaranteed to be real and positive. This satisfies all the conditions of Theorem 27 which completes the proof.
Corollary 29.
Assume that (99) holds and furthermore α2<1<α1b+α2 and c2>c¯. Then system (102) and (103) has a cycle of period two in (0,∞)2.
Figure 1 shows two orbits of system (102) and (103) from initial points (x0,y0)=(2.3,1) and (x0,y0)=(0.0001,0.0001). Although both orbits converge to the period two cycle, a shadow of the stable manifold of the fixed point is also seen in the initial segments of the two orbits. If the initial points start exactly on the stable manifold of x¯ then the solutions converge to x¯.
Orbits illustrating period two oscillations and the saddle point.
We studied the dynamics of a general planar system that includes many common stage-structured population models that evolve in discrete time. Our hypotheses regarding system (2a) and (2b) and its parameters are more general than what is typically assumed in population models with the aim of gaining a broader understanding of the mathematical properties of the system. The study in this paper is rigorous but incomplete and many issues remain. Generalizing the results in Section 4 to a level closer to that in Section 3 leads to a more comprehensive treatment of planar or two-stage, discrete population models. Among other things, this involves a consideration of systems involving the Ricker function where it is necessary to add the possibility of complex behavior. A resolution of these and related issues is left to future studies of system (2a) and (2b).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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