Influence of stochastic perturbation on prey–predator systems (original) (raw)
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A simple stochastic description of a model of a predator-prey system is given. The evolution of the system is described by means of It6's stochastic differential equations (SDEs), which are the natural stochastic generalization of the Lotka-Volterra deterministic differential equations. Since these SDEs do not satisfy the usual conditions for the existence and uniqueness of the solution, we state a theorem of existence; moreover we study the stability of the equilibrium point and perform a computer simulation to study the behaviour of the trajectories of solutions with given initial data and to estimate first and second moments.
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We analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.
In this paper, we present new results on deterministic sudden changes and stochastic fluctuations' effects on the dynamics of a two-predator one-prey model. We purpose to study the dynamics of the model with some impacting factors as the problem statement. e methodology depends on investigating the seasonality and stochastic terms which make the predator-prey interactions more realistic. A theoretical analysis is introduced for studying the effects of sudden deterministic changes, using three different cases of sudden changes. We show that the system in a good situation presents persistence dynamics only as a stable dynamical behavior. However, the system in a bad situation leads to three main outcomes as follows: first, constancy at the initial conditions of the prey and predators; second, extinction of the whole system; third, extinction of both predators, resulting in the growth of the prey population until it reaches a peak carrying capacity. We perform numerical simulations to study effects of stochastic fluctuations, which show that noise strength leads to an increase in the oscillations in the dynamical behavior and became more complex and finally leads to extinction when the strength of the noise is high. e random noises transfer the dynamical behavior from the equilibrium case to the oscillation case, which describes some unstable environments.
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We consider a system of stochastic equations which models the population dynamics of a prey-predator type. We show that the distributions of the solutions of this system are absolutely continuous. We analyse long-time behaviour of densities of the distributions of the solutions. We prove that the densities can converge in L 1 to an invariant density or can converge weakly to a singular measure.
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This paper investigates the deterministic and stochastic fluctuations of a predator-prey model. The predator is experienced in hunting two different prey simultaneously. Each prey has logistic growth in the absence of the predator. The rate of experience of the predator in hunting each prey is varied using a simulated dataset. The deterministic and stochastic nature of the dynamics of the system are investigated. Stability analysis is performed, using slight perturbation around the non-zero, interior equilibrium point, to determine where the system loses stability. The variation of the predatory experience parameter causes the system to experience Hopf bifurcations. These stability changes and the addition of stochastic noise are explored using time series graphs. The coexistence and extinction of the populations are affected over time .
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The investigation of interacting population models has long been and will continue to be one of the dominant subjects in mathematical ecology; moreover, the persistence and extinction of these models is one of the most interesting and important topics, because it provides insight into their behavior. The mean extinction-time depends on the initial population size and satisfies the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients; hence, finding analytical solutions poses severe problems, except in a few simple cases, so we can only compute numerical approximations (an idea already mentioned in Sharp and Allen (1998) [1]).
Dynamics of a stochastic Lotka–Volterra model perturbed by white noise
Journal of Mathematical Analysis and Applications, 2006
This paper continues the study of Mao et al. investigating two aspects of the equation dx(t) = diag x 1 (t),. .. , x n (t) b + Ax(t) dt + σ x(t) dW (t) , t 0. The first of these is to slightly improve results in [X. Mao, S. Sabais, E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal. 287 (2003) 141-156] concerning with the upper-growth rate of the total quantity n i=1 x i (t) of species by weakening hypotheses posed on the coefficients of the equation. The second aspect is to investigate the lower-growth rate of the positive solutions. By using Lyapunov function technique and using a changing time method, we prove that the total quantity n i=1 x i (t) always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower-growth rate.
Deterministic Chaos Versus Stochastic Oscillation in a Prey-Predator-Top Predator Model
Mathematical Modelling and Analysis, 2011
The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stabil...
Long-time behavior of a nonautonomous stochastic predator–prey model with jumps
Modern stochastics: theory and applications, 2021
It is proved the existence and uniqueness of the global positive solution to the system of stochastic differential equations describing a non-autonomous stochastic predator-prey model with modified version of Leslie-Gower term and Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises. We obtain sufficient conditions of stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak persistence in the mean and extinction of the solution to the considered system.
Lotka-Volterra predator-prey model with periodically varying carrying capacity
Physical Review E
We study the stochastic spatial Lotka-Volterra model for predator-prey interaction subject to a periodically varying carrying capacity. The Lotka-Volterra model with on-site lattice occupation restrictions (i.e., finite local carrying capacity) that represent finite food resources for the prey population exhibits a continuous active-toabsorbing phase transition. The active phase is sustained by the existence of spatiotemporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis in order to specifically delineate the impact of stochastic fluctuations and spatial correlations. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The (quasi-)stationary regime of our periodically varying Lotka-Volterra predator-prey system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity-switching environments, the mean-field rate equations predict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast and slow periodic switching. The analysis of the mean-field equations in the fast-switching regime enables a semiquantitative description of the (quasi-)stationary state.