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

• * Corresponding author
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).
• 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.

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

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

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

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

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

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

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

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

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

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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