# On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

## Abstract

We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka--Volterra system for an exponential random time after which it switches to a different deterministic Lotka--Volterra system. This switching procedure is then repeated. The resulting process is a piecewise deterministic Markov process (PDMP). In the case when the equilibrium points of the two deterministic Lotka--Volterra systems coincide we show that almost surely the trajectory does not converge to the common deterministic equilibrium. Instead, with probability one, the densities of the prey and the predator oscillate between 0 and $\infty$. This proves a conjecture of Takeuchi, Du, Hieu, and Sato [J. Math. Anal. Appl., 323 (2006), pp. 938--957]. The proof of the conjecture is a corollary of a result we prove about linear switched systems. Assume $(Y_t, I_t)$ is a PDMP that evolves according to $\frac{dY_t}{dt}=A_{I_t} Y_t$, where $A_0,A_1$ are $2\times2$ matrices and $I_t$ is a Markov chain on $\{0,1\}$ with transition rates $k_0,k_1>0$. If the matrices $A_0$ and $A_1$ are not proportional and are of the form $A_i := (\begin{smallmatrix} \alpha_i & \beta_i \\ \gamma_i & -\alpha_i \end{smallmatrix})$ with $\alpha_i^2 + \beta_i \gamma_i < 0$, then there exists $\lambda >0$ such that $\lim_{t \to \infty} \frac{\log \| Y_t \|}{t} = \lambda.$

## References

1.
L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer-Verlag, Berlin, 1998.
2.
L. Arnold, W. Horsthemke, and J. W. Stucki, The influence of external real and white noise on the Lotka-Volterra model, Biom. Journal, 21 (1979), pp. 451--471.
3.
Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), pp. 2937--2952.
4.
M. Benaïm, J. Hofbauer, and W. H. Sandholm, Robust permanence and impermanence for stochastic replicator dynamics, J. Biol. Dyn., 2 (2008), pp. 180--195.
5.
M. Benaïm, S. Le Borgne, F. Malrieu, and P.-A. Zitt, On the stability of planar randomly switched systems, Ann. Appl. Probab., 24 (2014), pp. 292--311.
6.
M. Benaïm, S. Le Borgne, F. Malrieu, and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), pp. 1040--1075.
7.
M. Benaïm and C. Lobry, Lotka Volterra in fluctuating environment or “How switching between beneficial environments can make survival harder,'' Ann. Appl. Probab., 26 (2016), pp. 3754--3785.
8.
M. Benaïm and S. J. Schreiber, Persistence of structured populations in random environments, Theor. Popul. Biol., 76 (2009), pp. 19--34.
9.
M. Benaïm and E. Strickler, Random switching between vector fields having a common zero, Ann. Appl. Probab., 29 (2019), pp. 326--375.
10.
J. Blath, A. Etheridge, and M. Meredith, Coexistence in locally regulated competing populations and survival of branching annihilating random walk, Ann. Appl. Probab., 17 (2007), pp. 1474--1507.
11.
P. Bougerol, Comparaison des exposants de Lyapounov des processus Markoviens multiplicatifs, Ann. Inst. Henri Poincaré Probab. Statist., 24 (1988), pp. 439--489.
12.
P. Cattiaux and S. Méléard, Competitive or weak cooperative stochastic Lotka--Volterra systems conditioned on non-extinction, J. Math. Biol., 60 (2010), pp. 797--829.
13.
P. Chesson, General theory of competitive coexistence in spatially-varying environments, Theor. Popul. Biol., 58 (2000), pp. 211--237.
14.
F. Colonius and G. Mazanti, Decay rates for stabilization of linear continuous-time systems with random switching, Math. Control Relat. Fields, 9 (2019), pp. 29--58.
15.
M. Costa, A piecewise deterministic model for a prey-predator community, Ann. Appl. Probab., 26 (2016), pp. 3491--3530.
16.
M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, J. R. Stat. Soc. Ser. B Stat. Methodol., 46 (1984), pp. 353--388.
17.
S. N. Evans, A. Hening, and S. J. Schreiber, Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments, J. Math. Biol., 71 (2015), pp. 325--359.
18.
S. N. Evans, P. L. Ralph, S. J. Schreiber, and A. Sen, Stochastic population growth in spatially heterogeneous environments, J. Math. Biol., 66 (2013), pp. 423--476.
19.
T. C. Gard and T. G. Hallam, Persistence in food webs. I. Lotka-Volterra food chains, Bull. Math. Biol., 41 (1979), pp. 877--891.
20.
M. E. Gilpin, Group Selection in Predator-Prey Communities, Princeton University Press, Princeton, NJ, 1975.
21.
R. Z. Has'minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Veroyatn. Primen., 5 (1960), pp. 196--214.
22.
A. Hening and D. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28, pp. 1893--1942.
23.
A. Hening and D. Nguyen, Persistence in stochastic Lotka-Volterra food chains with intraspecific competition, Bull. Math. Biol., 80 (2018), pp. 2527--2560.
24.
A. Hening and D. Nguyen, Stochastic Lotka-Volterra food chains, J. Math. Biol., 77 (2018), pp. 135--163.
25.
A. Hening, D. Nguyen, and G. Yin, Stochastic population growth in spatially heterogeneous environments: The density-dependent case, J. Math. Biol., 76 (2018), pp. 697--754.
26.
M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.
27.
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
28.
R. Z. Khasminskii and F. C. Klebaner, Long term behavior of solutions of the Lotka-Volterra system under small random perturbations, Ann. Appl. Probab., (2001), pp. 952--963.
29.
G. Lagasquie, A Note on Simple Randomly Switched Linear Systems, 2016, https://arxiv.org/abs/1612.01861.
30.
R. Lande, S. Engen, and B.-E. Saether, Stochastic Population Dynamics in Ecology and Conservation, Oxford University Press, Oxford, UK, 2003.
31.
S. D. Lawley, J. C. Mattingly, and M. C. Reed, Sensitivity to switching rates in stochastically switched ODEs, Commun. Math. Sci., 12 (2014), pp. 1343--1352.
32.
A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Company, Baltimore, MD, 1925.
33.
F. Malrieu and T. Hoa Phu, Lotka-Volterra with Randomly Fluctuating Environments: A Full Description, arXiv e-print, arXiv:1607.04395, 2016.
34.
N. Massarelli, K. Hoffman, and J. P. Previte, Effect of parity on productivity and sustainability of Lotka-Volterra food chains: Bounded orbits in food chains, J. Math. Biol., 69 (2014), pp. 1609--1626.
35.
R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl., 108 (2003), pp. 93--107.
36.
S. J. Schreiber, M. Benaïm, and K. A. S. Atchadé, Persistence in fluctuating environments, J. Math. Biol., 62 (2011), pp. 655--683.
37.
S. J. Schreiber and J. O. Lloyd-Smith, Invasion dynamics in spatially heterogeneous environments, Amer. Naturalist, 174 (2009), pp. 490--505.
38.
Y. Takeuchi, N. H. Du, N. T. Hieu, and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), pp. 938--957.
39.
V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Conseil, 3 (1928), pp. 3--51.

## Information & Authors

### Information

#### Published In

SIAM Journal on Mathematical Analysis
Pages: 3625 - 3640
ISSN (online): 1095-7154

#### History

Submitted: 22 June 2018
Accepted: 8 July 2019
Published online: 5 September 2019

### Authors

#### Funding Information

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung https://doi.org/10.13039/501100001711 : 2000020-149871/1

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