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

• * Corresponding author: Frithjof Lutscher
• 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.

Mathematics Subject Classification: 92D40, 34A37, 34A05, 34D05.

 Citation: • • Figure 1.  Illustration of the unique positive impulsive periodic orbit, corresponding to the unique positive steady state of the discrete map. Parameter values are: $\mu_1 = 0.7,\; \mu_2 = 0.3813,\; \eta_1 = 10,\; \eta_2 = 7.5811,\; \varepsilon_1 = \varepsilon_2 = 0.8711,\; \xi_1 = \xi_2 = 0.2941,\; \alpha = 5.3063,\; \rho = 0.8324$

Figure 2.  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

Figure 3.  The stability region of the semi-trivial equilibrium in the $u$-$v$ plane. Parameters are $\varepsilon_1 = 0.5307,\; \xi_1 = 0.2941,\; \alpha = 20.3063,\; \rho = 0.8324.$

Figure 4.  Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 2, except $\varepsilon_1 = \varepsilon_2 = 0.5307$

Figure 5.  Orbit diagram corresponding to fixing $\eta_1 = 10$, see the dashed line in Figure 4. Other parameters are as in Figure 4

Figure 6.  Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 4, except $\alpha = 20.3063$

Figure 7.  Orbit diagram corresponding to fixing $\eta_1 = 30$, see the dashed line in Figure 5. Other parameters are as in Figure 5

Figure 8.  Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 4, except $\varepsilon_1 = \varepsilon_2 = 0.2307$

Figure 9.  Orbit diagram corresponding to fixing $\eta_1 = 10$, see the dashed line in Figure 6. Other parameters are as in Figure 6

Figure 10.  Three species densities with different summer length. Parameters are: $m_1 = 0.6,\; m_2 = 0.3813,\; \theta_1 = 2,\; \theta_2 = 1.516,\; \delta_1 = \delta_2 = 0.5307,\; \xi_1 = \xi_2 = 0.2941,\; r = 5.3063,\; \rho = 0.8324,\; K = 5,\; a_1 = a_2 = 1$

Figure 11.  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

Figure 12.  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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