Preventing Noise-Induced Extinction in Discrete Population Models

Aproblemof the analysis and prevention of noise-induced extinction in nonlinear populationmodels is considered. For the solution of this problem, we suggest a general approach based on the stochastic sensitivity analysis. To prevent the noise-induced extinction, we construct feedback regulators which provide a low stochastic sensitivity and keep the system close to the safe equilibrium regime. For the demonstration of this approach, we apply ourmathematical technique to the conceptual but quite representative Ricker-type models. A variant of the Ricker model with delay is studied along with the classic widely used one-dimensional system.


Introduction
Population models, even in the deterministic case, demonstrate a wide variety of dynamic regimes, both equilibrium and oscillatory.These regular and chaotic regimes can be changed in consequence of the different-type bifurcations [1,2].Any living system is subject to the inevitable noise which can significantly complicate system's dynamics and cause abrupt ecological shifts and catastrophes [3][4][5].
In the population systems, an investigation of the underlying reasons of the transformation from the persistence to extinction is a challenging problem.Here, an analysis of the noise-induced extinction attracts the attention of many researchers [6][7][8][9][10][11][12].In the modern population dynamics, along with the problem of the analysis of unwanted shifts caused by noise, control problems for ecosystems are also highly important.
Stable regimes in the population dynamics are provided by intrinsic natural feedback mechanisms of the regulation.If such mechanisms are destroyed under various disturbances, then this desired stabilization can be achieved by the control based on additional artificial feedback [13][14][15][16][17].
Controlling nonlinear systems with regular and chaotic oscillations is an actively developed research domain.Following the pioneering work of Ott et al. [18], many researchers are involved in the investigations concerning the suppression of chaos to equilibria or periodic orbits [19][20][21].
The presence of stochastic disturbances significantly complicates the solution of control problems [22].In nonlinear dynamic systems, even with simple regular attractors, a small noise can cause unexpected and undesired noise-induced transitions.The underlying reason of such transitions is the high stochastic sensitivity of the initial deterministic attractors [23,24].A control approach based on the idea of reducing the stochastic sensitivity was developed in [25,26].
The discrete-time population models, even in the onedimensional case, are quite representative in the simulation of both regular and chaotic behavior.One of them is the well-known Ricker system [27].A theoretical analysis of the stochastic Ricker system in terms of Markov chains was carried out in [28,29].In this paper, we use the Ricker system as a conceptual model to study the probabilistic mechanisms of phenomena of the noise-induced extinction and contraction of the persistence zone.
In Section 2, we consider the one-dimensional stochastic Ricker system as a primary model.The main features of the uncontrolled model such as noise-induced extinction and contraction of the persistence zone are discussed.To prevent the extinction, we use a control approach oriented on the synthesis of the stable equilibrium with the small stochastic sensitivity.The mathematical background of this approach is briefly presented in the Appendix.
In Section 3, we study the Ricker system with delay as a secondary model.Here, we demonstrate how the mathematical theory of the stochastic sensitivity synthesis can be applied to the systems of higher dimensions.

Controlling Stochastic Sensitivity in 1D Ricker Model
Consider the deterministic Ricker model [27] where  is the size of the population and  > 0 is a parameter of the intrinsic growth rate.System (1) has two equilibria:  0 = 0 and  1 = 1.The equilibrium  0 is unstable for any  > 0, and the equilibrium  1 is stable for 0 <  < 2. For  > 2, the system performs Feigenbaum's scenario of perioddoubling bifurcations and alternations of regular and chaotic zones.The attractors of the deterministic model (1) are shown in Figure 1.Note that for any  the population exists in various forms, both regular and chaotic, with the unlimited growth of the amplitude under increasing .So, in the framework of this deterministic model, the persistence -zone is unbounded: It can be underlined that minimum values of the population size in oscillatory modes tend to zero as  increases.In the presence of random disturbances, such proximity of the population size to zero can cause noise-induced extinction.To study this phenomenon, we consider the following stochastic variant of the Ricker model: where   is a standard uncorrelated Gaussian random process with parameters (  ) = 0 and ( 2  ) = 1 and  is the noise intensity.Here, the environmental noise scales proportionally with the population size.This model takes into account the biological fact that the population cannot withstand the negative environmental disturbance that is more than its own size.Mathematically, this obvious feature of the interplay of the population and environment is modeled in (2) by the function In the framework of this model, the population size  is nonnegative even for large values of random disturbances.Consider how noise can destroy the persistence regime of the population system.For small noise, random trajectories slightly deviate from the initial deterministic attractor (see blue curves in Figure 2).With increasing noise, these deviations increase too, and the random trajectory can intersect  =  0 and further be zero for any  (see red curves in Figure 2).Such annihilation can be interpreted as the noiseinduced extinction of the population.Note that, for larger , the noise with smaller intensity  causes the noise-induced extinction.It can be clearly seen in Figure 3 where random states of system (2) are shown for three values of .For any , we started from the deterministic attractor and plotted 100 iterations of the stochastic system (2) after the transient 100 steps.Here, it can be seen that the increasing noise contracts the persistence zone.For instance, in the parametric zone where the deterministic system (1) exhibits persistence with chaotic oscillations, the stochastic system (2) with the noise intensity  = 0.1 demonstrates total extinction.
Consider now how to avoid this noise-induced extinction using the appropriate control procedures.Here, our aim is to stabilize the equilibrium  1 in a wide range of the parameter  > 0 and provide small-amplitude stochastic oscillations near  1 .
Consider the stochastic model with control input where   is formed by the feedback regulator To stabilize the stochastic oscillations of system (4) with regulator (5), we will use a theory of the stochastic sensitivity synthesis presented in the Appendix.Here, The set of the feedback coefficients  of regulator (5) stabilizing  1 = 1 of the deterministic system is given by the inequality  − 2 <  < .Following (A.14), we have where   is the stochastic sensitivity of the equilibrium  1 = 1 in stochastic model (4) with regulator (5).In absence of the control ( = 0), the stochastic sensitivity (see the blue curve in Figure 4) of  1 in the zone 0 <  < 2 of its stability is given by the formula Formally, for  ≥ 2,  0 is equal to infinity.A problem of the synthesis of the assigned stochastic sensitivity  is solved by regulator (5) with the feedback coefficient (see (A.13)) Here, the attainability set of the stochastic sensitivity is defined by a simple inequality  ≥ 1. Results of the modeling of system (2) without control and system (4) and ( 5) with control which synthesizes different values of the stochastic sensitivity  are shown in Figure 5.In Figure 5(a), random states are plotted for  = 0.01 with control providing  = 10 (green points) and  = 1 (red points).As can be seen, the regulator stabilizes  1 = 1 and provides uniform dispersion in a wide range of the parameter .A value of the dispersion can be controlled by the assigned .Here, the value  = 1 is the minimum attainable.In Figure 5(b), random states are plotted for larger noise intensity  = 0.1 with control providing this minimum value  = 1.
In Figure 6, time series of the stochastic system with  = 2.5 and  = 0.01 without control are plotted by a grey color and those with control are plotted by blue for regulators providing  = 1 (a) and  = 10 (b).Here, the control is switched on at  = 20.
Thus, our approach, based on the synthesis of the regulator minimizing the stochastic sensitivity, provides structural stabilization, decreases the amplitude of oscillations, and prevents the noise-induced extinction in the population system.Note that these results can be also interpreted as the suppression of chaos in the corresponding -zone.
Further, we will show how this approach can be applied to the population system of higher dimension.In the next section, we demonstrate the effectiveness of our control approach on the example of the Ricker model with delay.

Stabilization of the Stochastic Ricker Model with Delay
Consider the following modification of the Ricker model ( 1) with delay: The Ricker model with delay was studied in [30,31].Equation ( 10) can be rewritten as the following system: This system has two equilibria:  0 (0, 0) and  1 (1, 1).For any  > 0, the equilibrium  0 is unstable.The equilibrium  1 is stable for 0 <  < 1.When the parameter  passes the value  = 1, the equilibrium  1 becomes unstable, and system (11) exhibits the Neimark-Sacker bifurcation [32] with the birth of a closed invariant curve.In Figure 7, -coordinates of attractors of model (11) are plotted.With increasing , in system (11), an alternation of regular and chaotic oscillations with increasing amplitude is observed.Here, similar to onedimensional Ricker model ( 1), the persistence -zone is unbounded: 0 <  < ∞.
Consider now the stochastic variant of the Ricker model with delay: Here,   is a standard uncorrelated Gaussian random process with parameters (  ) = 0 and ( 2  ) = 1, and  is the noise intensity.
For this model with delay, the noise contracts the persistence zone as well.In the presence of noise, this persistence -zone becomes finite: the larger the noise intensity , the smaller the size of the persistence zone.It is clearly seen in Figure 8 where random states of system (12) are shown for three values of .Consider now how to stabilize the equilibrium  1 for the whole interval  > 0 and provide small-amplitude stochastic oscillations near this equilibrium.
Consider the stochastic model with control input where   is formed by the feedback regulator Here,  1 = 1 and  1 = 1 are coordinates of the equilibrium  1 .
In notations of the Appendix, for system (13) and the equilibrium  1 , we have the following: The stability condition ( + ) < 1 is equivalent to the system of inequalities: || < 1, || < 1 − .So, parameters  1 and  2 of the stabilizing regulator (14) are Here, due to the symmetry of the matrix , we have  21 =  12 .System (17) has the solution In the absence of the control ( 1 =  2 = 0), we have The function  11 () is plotted in the stability zone 0 <  < 1 of the equilibrium  1 in Figure 9.It can be noted that the introduced delay increases the stochastic sensitivity of the stable equilibrium (compare Figures 4 and 9).The problem of the synthesis of the assigned stochastic sensitivity matrix  is reduced to finding  and  from system (17).The corresponding solution is given by the formulas ) .
(20) Note that this solution cannot be found for any matrix .Elements of the attainable matrices satisfy the following conditions: The attainability domain  is plotted in Figure 10.Any attainable matrix can be provided by regulator (14) with parameters As one can see, in the attainability domain, the minimum value  11 = 1 corresponds to  12 = 0. So, we get the stochastic matrix  = [ 1 0 0 1 ] and  1 = −1,  2 = .Results of the modeling of system (12) without control and system (13) with regulator (14) which synthesizes minimum values of the attainable stochastic sensitivity  11 = 1 are shown in Figures 11 and 12.In Figure 11(a), for  = 0.01, random states of the uncontrolled system are plotted by blue color, and for the system with control they are shown by red.In Figure 11(b), for larger noise intensity  = 0.1, random states of the uncontrolled system are plotted by light blue, and for the system with control they are shown by red.As can be seen, the regulator stabilizes  1 and provides uniform dispersion in a wide range of the parameter .
In Figure 12, phase trajectories and time series of the coordinate of the stochastic system with  = 1.2 without control are plotted by blue color, and with control they are plotted by red color for the optimal regulator providing  11 = 1.Here, we consider  = 0.01 (a) and  = 0.1 (b).For small noise ( = 0.01), our regulator sharply decreases the amplitude of stochastic oscillations.For larger noise ( = 0.1), in the system without control, the population becomes extinct.The constructed regulator prevents this unwanted ecological shift, stabilizes  1 , and provides admissible stochastic oscillations near  1 .

Conclusions
We studied the noise-induced extinction in nonlinear population models.For the mathematical analysis of this phenomenon, we used a general theoretical approach based on the stochastic sensitivity functions technique.Constructive abilities of this approach were demonstrated on the conceptual 1D and 2D stochastic Ricker models.To prevent the noise-induced extinction, we reduced a level of the stochastic sensitivity of the nontrivial equilibrium by the appropriate control.A mathematical background of the theory of the stochastic sensitivity synthesis by feedback regulators was briefly presented.For Ricker models, it was shown how these regulators provide a low stochastic sensitivity and keep the system close to the required safe equilibrium regime.

Figure 1 :
Figure 1: Bifurcation diagram of the deterministic system (1).The unstable equilibrium  0 is shown by the red dashed line.

Figure 7 :
Figure 7: Bifurcation diagram of the deterministic system(11).The unstable equilibrium  0 is shown by the red dashed line.