The Effect of Prey Refuges on a Three Species Food Chain Model

Survival analysis of a stochastic predator–prey model with prey refuge and fear effect

“…and assume that the initial value of model (2) is (x(0), y(0)) = (0.6, 0.5). First, in order to investigate the impact of white noise on the dynamics of model (2), we choose all parameter values in (23) and m = 0.1, K = 0.3, σ 1 = σ 2 = 0.01. By calculating, we get from (6) that c 2 > 0.148.…”

“…Next, in order to illustrate the influence of fear effect to model (2) through numerical simulation, we choose different values of K, say K = 0, K = 0.1 and K = 1, σ 1 = σ 2 = 0.01. For the remaining parameter values, we keep them the same as in (23). We can compute that these parameters satisfy Assumptions (H 1 ) and (H 2 ) and therefore model (2) has a stationary distribution, which means that both prey and predator populations are persistent.…”

“…Finally, we numerically illustrate the impact of a prey refuge to model (2) and choose different values of m, say m = 0, m = 0.2, m = 0.5 and m = 0.7, σ 1 = σ 2 = 0.01. For the remaining parameter values, we keep them the same as in (23). We can easily compute that these parameters satisfy assumptions (H 1 ) and (H 2 ).…”

“…This indicates that a prey refuge can protect and prevent the extinction of preys, and thus, it plays an important role in the interaction between predator and prey populations. Recently, the research of the predator-prey models with prey refuges has attracted many mathematicians' attention and some interesting results have been obtained [19,23,25,32].…”

“…Numerical simulation for model(1) and model (2) with initial value (x(0), y(0)) = (0.6, 0.5) and different K, respectively. The parameters are taken as(23) and m = 0.1, σ 1 = σ 2 = 0.01.…”