Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response

Guohong Zhang; Xiaoli Wang

doi:10.3934/dcdsb.2018076

Article Contents

2018, Volume 23, Issue 9: 3755-3786. Doi: 10.3934/dcdsb.2018076

This issue Previous Article Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations Next Article Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response

Guohong Zhang and
Xiaoli Wang^,

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author
* Corresponding author

Received: December 31, 2015

Revised: August 31, 2017

Early access: March 2018

Published: September 2018

This work is supported by National Science Foundation of China(11701472), Fundamental Research Funds for the Central Universities(XDJK2016C121, XDJK2017C055), the Ph.D. Foundation of Southwest University (SWU112099, SWU116069).

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Abstract

Extinction and coexistence of species are two fundamental issues in systems with IGP. In this paper, we constructed a mathematical model with IGP by introducing heterogeneous environment and B-D functional response between the predator and prey. First, some sufficient conditions for the extinction and permanence of the time-dependent system were obtained by using comparison principle and upper and lower solution method. Second, we got some necessary and sufficient conditions for the existence of coexistence states by means of the fixed point index theory. In addition, we discussed the uniqueness and stability of coexistence state under some conditions. Finally, we studied the effects of the parameters in system on the spatial distribution of species and obtained some interesting results about the extinction and coexistence of species by using numerical simulations.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 92D15, 92D25.

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Figure 1. Coexistence induced by $r_{1}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 3$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{2} = 0.3$, $r_{3} = 0.2$, $e_{1} = e_{2} = e_{3} = 1$

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Figure 2. Coexistence induced by $r_{2}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{3} = 0.2$, $e_{1} = e_{2} = e_{3} = 1$

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Figure 3. Coexistence and extinction induced by $r_{3}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $e_{1} = e_{2} = e_{3} = 1$

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Figure 4. Coexistence induced by interference of IGpredators ($e_{2}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = 0.2, e_{1} = e_{3} = 1$

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Figure 5. Coexistence induced by interference among IGpredators ($e_{3}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = 0.2$, $e_{1} = e_{2} = 1$

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Figure 6. Extinction of IGprey driven by interference among IGprey ($e_{1}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = 0.2$, $e_{2} = e_{3} = 1$

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Figure 7. Extinction of IGpredator driven by $e_{2}$ and $e_{3}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = -0.5$, $e_{1} = 1$

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Figure 8. Coexistence induced by IGP ($a_{2}$ and $m_{2}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 0.5$, $m_{1} = 2.5$, $m_{3} = 0.25$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.1$, $r_{3} = -0.15$, $e_{1} = e_{2} = e_{3} = 10$

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Figure 9. Extinction of IGprey driven by IGP ($a_{2}$ and $m_{2}$) with fixed parameter values $a_{1} = 1$, $a_{3} = 0.5$, $m_{1} = 0.5$, $m_{3} = 0.25$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.1$, $r_{3} = -0.15$, $e_{1} = e_{2} = e_{3} = 1$

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