# American Institute of Mathematical Sciences

March  2020, 25(3): 877-901. doi: 10.3934/dcdsb.2019194

## Global dynamics of a reaction-diffusion system with intraguild predation and internal storage

 1 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China 2 Department of Mathematics and National Center of Theoretical Science, National Tsing-Hua University, Hsinchu 300, Taiwan 3 Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan 4 Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

* Corresponding author: fbwang@mail.cgu.edu.tw, fbwang0229@gmail.com

Received  November 2018 Revised  March 2019 Published  March 2020 Early access  September 2019

Fund Project: The first author is supported by NSF of China (11671243, 11572180), the Fundamental Research Funds for the Central Universities (GK201701001).

This paper presents a reaction-diffusion system modeling interactions of the intraguild predator and prey in an unstirred chemostat, in which the predator can also compete with its prey for one single nutrient resource that can be stored within individuals. Under suitable conditions, we first show that there are at least three steady-state solutions for the full system, a trivial steady-state solution with neither species present, and two semitrivial steady-state solutions with just one of the species. Then we establish that coexistence of the intraguild predator and prey can occur if both of the semitrivial steady-state solutions are invasible by the missing species. Comparing with the system without predation, our numerical simulations show that the introduction of predation in an ecosystem can enhance the coexistence of species. Our mathematical arguments also work for the linear food chain model (top-down predation), in which the top-down predator only feeds on the prey but does not compete for nutrient resource with the prey. In our numerical studies, we also do a comparison of intraguild predation and top-down predation.

Citation: Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194
##### References:

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##### References:
The effects of the nutrient supply concentration $R^{(0)}$: (A, C) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$, and (B, D) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$. $R^{(0)} = 1.5\times10^{-5}\, \mathrm{mol}\,\mathrm{l}^{-1}$ in (A, B), and $R^{(0)} = 2.5\times10^{-5}\, \mathrm{mol}\,\mathrm{l}^{-1}$ in (C, D)
Bifurcation diagrams of positive steady state solutions to (5)-(7) with the bifurcation parameter $R^{(0)}$ ranging from $0.5\times10^{-5}$ to $2.0\times10^{-4}\, \mathrm{mol}\,\mathrm{l}^{-1}.$ (A) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$, and (B) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$
The effects of the diffusion rate $d$: (A, C, E) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$, and (B, D, F) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$. $d = 0.08\, \mathrm{day}^{-1}$ in (A, B), $d = 0.12\, \mathrm{day}^{-1}$ in (C, D), and $d = 0.16\, \mathrm{day}^{-1}$ in (E, F)
Bifurcation diagrams of positive steady state solutions to (5)-(7) with the bifurcation parameter $g_{\max}$ ranging from $0$ to $120\ \mathrm{cells}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$. (A) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$, and (B) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$
Common parameters used in intraguild predation and top-down predation
 Quantity Value Quantity Value $\gamma$ $10\, \mathrm{day}^{-1}$ $a_{\max,1}$ $12.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$ $K_1$ $9.0\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1}$ $K_2$ $6.5\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1}$ $\mu_{\max,1}$ $0.7\,\mathrm{day}^{-1}$ $\mu_{\max,2}$ $2.2\,\mathrm{day}^{-1}$ $Q_{\min,1}$ $2.6\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $Q_{\min,2}$ $1.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $Q_{\max,1}$ $9.5\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $Q_{\max,2}$ $32.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $b$ 2.37 $K_0$ $4.0\times10^{8}\,\mathrm{cells}\,\mathrm{l}^{-1}$
 Quantity Value Quantity Value $\gamma$ $10\, \mathrm{day}^{-1}$ $a_{\max,1}$ $12.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1}$ $K_1$ $9.0\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1}$ $K_2$ $6.5\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1}$ $\mu_{\max,1}$ $0.7\,\mathrm{day}^{-1}$ $\mu_{\max,2}$ $2.2\,\mathrm{day}^{-1}$ $Q_{\min,1}$ $2.6\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $Q_{\min,2}$ $1.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $Q_{\max,1}$ $9.5\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $Q_{\max,2}$ $32.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1}$ $b$ 2.37 $K_0$ $4.0\times10^{8}\,\mathrm{cells}\,\mathrm{l}^{-1}$
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