American Institute of Mathematical Sciences

Prey-predator model with nonlocal and global consumption in the prey dynamics

 1 Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India 2 Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France 3 INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France 4 Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

* Corresponding author: Vitaly Volpert

Received  January 2019 Published  November 2019

Fund Project: The third author is supported by "RUDN University Program 5-100".

A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.

Citation: Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020180
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Single pulse solution for prey and predator population for $b = 10, d_1 = 0.1, d_2 = 0.1$
Moving pulse for $b = 10, d_1 = 0.3, d_2 = 0.1$ (a) after time $t = 500$; (b) after time $t = 600$; (c) $x$-$t$ profile for $L = 20$
For two bifurcation diagrams, $a = 1$, $\sigma_1 = 0.1$, $k_1 = k_2 = 0.35$, $\sigma_2 = 0.2$ are fixed. (a) Bifurcation diagram in $(L,d_1)$-plane for the values of parameters $b = 5$, $\alpha = \beta = 0.35$, $d_2 = 0.1$. (b) Bifurcation diagram in $(d_1,b)$-plane for the values of other parameters $\alpha = \beta = 0.363$, $d_2 = 1$
Periodic travelling wave in the case of nonlocal consumption followed by a spatio-temporal structure. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $u(x,t)$. The values of other parameters are $a = 1,b = 1, \alpha = \beta = 0.363, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1$
Multiple moving pulses in the case of nonlocal consumption. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $u(x,t)$. The values of other parameters are $a = 1,b = 1, \alpha = \beta = 0.375, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1$
Different regimes observed in the case of nonlocal consumption presented on the $(d_1,N)$ parameter plane for $\alpha = 0.35$ (left) and $(N,\alpha)$ parameter plane for $d_1 = 1$ (right). The values of other parameters are $a = 1,b = 1,\sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_2 = 1$
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