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# On carrying-capacity construction, metapopulations and density-dependent mortality

• We present a mathematical model for competition between species that includes variable carrying capacity within the framework of niche construction. We make use the classical Lotka-Volterra system for species competition and introduce a new variable which contains the dynamics of the constructed niche. The paper illustrates that the total available patches at equilibrium always exceeds the constructedniche at equilibrium in the absence of species.

Mathematics Subject Classification: Primary:92D25, 92D40;Secondary:34D20, 37C75.

 Citation: • • Figure 2.  Possible scenarios for coexistence and extinction regions when $Q>1$. These are regions where an equilibrium point is feasible. The conditions that define them are necessary but not sufficient for their existence. Region e) is dashed to indicate that it is not feasible, given that not exist one or two species

Figure 1.  Qualitative behavior of $Q$ as a function of $(\kappa_2 ,\kappa_3 )\in [0,5]\times [0,5]$ for a) $\omega_2/\omega_1>1$, b) $\omega_2/\omega_1<1$. Note the narrowing of the range where $Q>1$ when going from a) to b).

Figure 6.  Possible scenarios for coexistence and extinction regions when $Q<1$. These are regions where an equilibrium point is feasible. The conditions that define them are necessary but not sufficient for their existence. Region e) is dashed to indicate that it is not feasible, given that not exist one or two species

Figure 3.  Coexistence of two species for $Q>1$. This scenario corresponds to the region b) of the Fig. 2. The parameters are $\kappa_1=1,$ $\kappa_2=3.5$, $\kappa_3=1.4$, $b=0.2$, $\beta_1=3.4$, $\beta_2=1.6$ $\sigma=0.1$, $p=0.5$, $d=1$, $e=1$, $u=0.18$, $c_1=0.9$, $c_2=0.2$

Figure 4.  Colonization by the specie $I_1$ for $Q>1$. This scenario corresponds to the region a) of the Fig. 2. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.4$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$

Figure 5.  Colonization by the specie $I_2$ for $Q>1$. This scenario corresponds to the region c) of the Fig. 2. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=3.4$ $\sigma=0.5$, $p=0.1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$

Figure 7.  Colonization by the specie $I_1$ for $Q<1$. This scenario corresponds to the region c) of the Fig. 6. The parameters are $\kappa_1=1,$ $\kappa_2=1.0$, $\kappa_3=0.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=0.1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.9$, $c_2=0.9$

Figure 8.  Colonization by the specie $I_1$ for $Q<1$. This scenario corresponds to the region c) of the Fig. 6. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=0.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$

Figure 9.  Colonization by the specie $I_2$ for $Q<1$. This scenario corresponds to the region a) of the Fig. 6. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.1$, $b=0.8$, $\beta_1=3.8$, $\beta_2=3.4$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$

Figure 10.  Eigenvalues region for non-colonized state

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