American Institute of Mathematical Sciences

February  2022, 15(2): 245-263. doi: 10.3934/dcdss.2020468

Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species

 Department of Mathematical Sciences, College of Science, UAE University, Al Ain, 15551, UAE

* Corresponding author: F.A. Rihan (frihan@uaeu.ac.ae)

Received  April 2020 Revised  September 2020 Published  February 2022 Early access  November 2020

Fund Project: This work supported by UPAR-Project (Code # G00003440)

Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter ${\mathcal R}_0^s$ that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.

Citation: Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2022, 15 (2) : 245-263. doi: 10.3934/dcdss.2020468
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Shows numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), when $\tau_1 = 1.25$, $\tau_2 = 0.6$ and $\tau_3 = 0.5$, with noise intensities $\sigma_1^2 = 0.08$, $\sigma_2^2 = 0.1$, $\sigma_3^2 = 0.06$, and parameter values: $r_1 = 0.2$, $r_2 = 0.6$, $K_1 = 0.7$, $K_2 = 0.8$, $\alpha_1 = 0.3$, $\alpha_2 = 0.6$, $\alpha_3 = 0.8$, $\beta = 0.1$, $\delta = 0.8$, $a_1 = 1$, $a_2 = 1.4$. For ${\mathcal R}_0^s>1$, the stochastic model has a unique ergodic stationary distribution $\pi(.)$ of stochastic system (2)
Numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), with parameter values given in Example 2, with noise intensities: $\sigma_1^2 = 0.03$, $\sigma_2^2 = 0.02$ and $\sigma_3^2 = 1.4$. When ${\mathcal R}_0^s<1$, we can clearly see that the predator goes to extinct
, but with noise intensities $\sigma_1^2 = 1.2$, $\sigma_2^2 = 1.2$ and $\sigma_3^2 = 0.5$. When $r_1<\frac{\sigma_1^2}{2}$, $r_2<\frac{\sigma_2^2}{2}$ and ${\mathcal R}_0^s<1$, we can clearly see that all the species goes to extinct. A strong intensity of noise can be a cause for extinction of the prey species, which will also drive predator population to extinct">Figure 3.  Shows numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), with the same parameter values of Figure 2, but with noise intensities $\sigma_1^2 = 1.2$, $\sigma_2^2 = 1.2$ and $\sigma_3^2 = 0.5$. When $r_1<\frac{\sigma_1^2}{2}$, $r_2<\frac{\sigma_2^2}{2}$ and ${\mathcal R}_0^s<1$, we can clearly see that all the species goes to extinct. A strong intensity of noise can be a cause for extinction of the prey species, which will also drive predator population to extinct
, but with $\tau_1 = 10$, $\tau_2 = 0.1$ and $\tau_3 = 0.1$, under the noise intensities $\sigma_1^2 = 0.2$, $\sigma_2^2 = 0.2$ and $\sigma_3^2 = 0.2$. Clearly, time-delays can lead to Hopf-type bifurcations of deterministic systems">Figure 4.  Numerical simulations of deterministic DDEs (1) (left) and SDDEs (2) (right), with parameter values of Figure 1, but with $\tau_1 = 10$, $\tau_2 = 0.1$ and $\tau_3 = 0.1$, under the noise intensities $\sigma_1^2 = 0.2$, $\sigma_2^2 = 0.2$ and $\sigma_3^2 = 0.2$. Clearly, time-delays can lead to Hopf-type bifurcations of deterministic systems
, there is an explosion of population with deterministic model (left); While the noise prevent such explosion of the population (right)">Figure 5.  Shows the effect of white noise to prevent the explosion of the population. When $\beta = 0.5$, with the same parameter values of Figure 1, there is an explosion of population with deterministic model (left); While the noise prevent such explosion of the population (right)
One biological meaning for the parameters of model (2)
 Parameters Description $r_1$, $r_2$ intrinsic growth rate for x and y $k_1$, $k_2$ carrying capacity for x and y $\alpha_1$, $\alpha_2$ rate of predation of preys x and y $\beta$ rate of cooperation of preys x and y against predator z $\delta$ Predator death rate $\alpha_3$ rate of intra-species competition within the predators $a_1$, $a_2$ transformation rate of predator to preys $x$ and $y$.
 Parameters Description $r_1$, $r_2$ intrinsic growth rate for x and y $k_1$, $k_2$ carrying capacity for x and y $\alpha_1$, $\alpha_2$ rate of predation of preys x and y $\beta$ rate of cooperation of preys x and y against predator z $\delta$ Predator death rate $\alpha_3$ rate of intra-species competition within the predators $a_1$, $a_2$ transformation rate of predator to preys $x$ and $y$.
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