# American Institute of Mathematical Sciences

January  2021, 26(1): 269-297. doi: 10.3934/dcdsb.2020140

## Coexistence of competing consumers on a single resource in a hybrid model

 1 Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China 2 Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N6N5, Canada 3 Department of Mathematics and Statistics, Department of Biology, University of Ottawa, Ottawa, K1N6N5, Canada

* Corresponding author: Frithjof Lutscher

Received  November 2019 Revised  February 2020 Published  April 2020

The question of whether and how two competing consumers can coexist on a single limiting resource has a long tradition in ecological theory. We build on a recent seasonal (hybrid) model for one consumer and one resource, and we extend it by introducing a second consumer. Consumers reproduce only once per year, the resource reproduces throughout the"summer" season. When we use linear consumer reproduction between years, we find explicit expressions for the trivial and semi-trivial equilibria, and we prove that there is no positive equilibrium generically. When we use non-linear consumer reproduction, we determine conditions for which both semi-trivial equilibria are unstable. We prove that a unique positive equilibrium exists in this case, and we find an explicit analytical expression for it. By linear analysis and numerical simulation, we find bifurcations from the stable equilibrium to population cycles that may appear through period-doubling or Hopf bifurcations. We interpret our results in terms of climate change that changes the length of the"summer" season.

Citation: Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140
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##### References:
Illustration of the unique positive impulsive periodic orbit, corresponding to the unique positive steady state of the discrete map. Parameter values are: $\mu_1 = 0.7,\; \mu_2 = 0.3813,\; \eta_1 = 10,\; \eta_2 = 7.5811,\; \varepsilon_1 = \varepsilon_2 = 0.8711,\; \xi_1 = \xi_2 = 0.2941,\; \alpha = 5.3063,\; \rho = 0.8324$
The existence range of the positive equilibrium with respect to parameters $\mu_1$ and $\eta_1$. The upper boundary curve corresponds to condition (39), while the lower curve corresponds to (45). Parameters are as in Figure 1, unless otherwise noted
The stability region of the semi-trivial equilibrium in the $u$-$v$ plane. Parameters are $\varepsilon_1 = 0.5307,\; \xi_1 = 0.2941,\; \alpha = 20.3063,\; \rho = 0.8324.$
Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 2, except $\varepsilon_1 = \varepsilon_2 = 0.5307$
Orbit diagram corresponding to fixing $\eta_1 = 10$, see the dashed line in Figure 4. Other parameters are as in Figure 4
Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 4, except $\alpha = 20.3063$
Orbit diagram corresponding to fixing $\eta_1 = 30$, see the dashed line in Figure 5. Other parameters are as in Figure 5
Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 4, except $\varepsilon_1 = \varepsilon_2 = 0.2307$
Orbit diagram corresponding to fixing $\eta_1 = 10$, see the dashed line in Figure 6. Other parameters are as in Figure 6
Three species densities with different summer length. Parameters are: $m_1 = 0.6,\; m_2 = 0.3813,\; \theta_1 = 2,\; \theta_2 = 1.516,\; \delta_1 = \delta_2 = 0.5307,\; \xi_1 = \xi_2 = 0.2941,\; r = 5.3063,\; \rho = 0.8324,\; K = 5,\; a_1 = a_2 = 1$
Three species densities with different summer length. $\rho,\; \xi_1,\; \xi_2$ increasing in $T$ with $\omega = 0.7$, other parameters are as in Figure 7, except $\delta_1 = \delta_2 = 0.0711$ in right plot
Three species coexist with linear reproduction. Parameters are $\mu_1 = 0.7,\; \mu_2 = 0.5378,\; \eta_1 = 5.9,\; \eta_2 = 7.2284,\; \xi_1 = 0.6681,\; \xi_2 = 0.1788,\; \alpha = 55.0495,\; \rho = 0.9599.$
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