# American Institute of Mathematical Sciences

January  2021, 26(1): 217-267. doi: 10.3934/dcdsb.2020328

## Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence

 1 Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089, USA 2 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA

* Corresponding author

Received  March 2020 Revised  October 2020 Published  November 2020

To underpin the concern that environmental change can flip an ecosystem from stable persistence to sudden total collapse, we consider a class of so-called ecoepidemic models, predator – prey/host – parasite systems, in which a base species is prey to a predator species and host to a micro-parasite species. Our model uses generalized frequency-dependent incidence for the disease transmission and mass action kinetics for predation.

We show that a large variety of dynamics can arise, ranging from dynamic persistence of all three species to either total ecosystem collapse caused by high transmissibility of the parasite on the one hand or to parasite extinction and prey-predator survival due to low parasite transmissibility on the other hand. We identify a threshold parameter (tipping number) for the transition of the ecosystem from uniform prey/host persistence to total extinction under suitable initial conditions.

Citation: Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328
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Summary of the dynamics of the host-parasite subsystem when $\xi$ is decreasing
 $^{{\dagger}}$ means that this event only occurs if the corresponding parameter inequality is strict or $\xi$ is strictly decreasing; GAS stands for "globally asymptotically stable." Parameter Values Dynamics ${{\sigma}}\le \frac{{{\mu}}}{h'(0)}$ $S(0)>0 \Longrightarrow r(t)\to 0, S(t)\to K$ $\frac{{{\mu}}}{h'(0)}<{{\sigma}}\le \frac{{{\mu}}+g(0)}{h'(0)}$ no $(0,r^\circ)$, $(S^*,r^*)$ GAS$^{{\dagger}}$ for $(0,\infty)^2$ $\frac{{{\mu}}+g(0)}{h'(0)}<{{\sigma}}<\frac{g(0)}{h(g(0)/{{\mu}})}$ $\exists (0,r^\circ)$, $(S^*,r^*)$ GAS for $(0,\infty)^2$ $\frac{g(0)}{h(g(0)/{{\mu}})} \le {{\sigma}}<\frac{{{\mu}}+g(0)}{h(\infty)}$ $\exists (0,r^\circ)$, $r(0)>0 \Longrightarrow S(t)\to 0$ $\frac{{{\mu}}+g(0)}{h(\infty)}\le {{\sigma}}$ $r(0)> 0 \Longrightarrow S(t)\to 0, (r(t)\to\infty)^{{\dagger}}$
 $^{{\dagger}}$ means that this event only occurs if the corresponding parameter inequality is strict or $\xi$ is strictly decreasing; GAS stands for "globally asymptotically stable." Parameter Values Dynamics ${{\sigma}}\le \frac{{{\mu}}}{h'(0)}$ $S(0)>0 \Longrightarrow r(t)\to 0, S(t)\to K$ $\frac{{{\mu}}}{h'(0)}<{{\sigma}}\le \frac{{{\mu}}+g(0)}{h'(0)}$ no $(0,r^\circ)$, $(S^*,r^*)$ GAS$^{{\dagger}}$ for $(0,\infty)^2$ $\frac{{{\mu}}+g(0)}{h'(0)}<{{\sigma}}<\frac{g(0)}{h(g(0)/{{\mu}})}$ $\exists (0,r^\circ)$, $(S^*,r^*)$ GAS for $(0,\infty)^2$ $\frac{g(0)}{h(g(0)/{{\mu}})} \le {{\sigma}}<\frac{{{\mu}}+g(0)}{h(\infty)}$ $\exists (0,r^\circ)$, $r(0)>0 \Longrightarrow S(t)\to 0$ $\frac{{{\mu}}+g(0)}{h(\infty)}\le {{\sigma}}$ $r(0)> 0 \Longrightarrow S(t)\to 0, (r(t)\to\infty)^{{\dagger}}$
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