| Abstract: |
| Leslie-Gower competitive systems in discrete time for $d$ species can be described as follows: For time $t\geq 0$ and $ i=1,\ldots ,d,$
\begin{equation}
x_i(t+1)=\frac{a_ix_i(t)}{1+\displaystyle\sum^n_{j=1}c_{ij}x_j(t)},
\end{equation}
where $x_i(t)$ means the population size of the $i$-th species at time $t$, all carrying capacities $(a_{i}-1)/c_{ii} >0$ and the inter-specific coefficients $c_{ij}>0$. By rescaling, we may assume wlog that all $c_{ii}=1$. Biologists believe that only the specifies with highest carrying capacity will survive as time goes to $\infty$. It is not always so as Cushing et ce. (2004) gave a complete description of the global behavior for $d=2$. Ackleh and Jang (2005) conjectured that such a principle holds when all $c_{ij}=1$. Chow and Hsieh (2013) verified this conjecture. A simple proof can be found in Ackleh , Sacker and Salceanu (2014). For two species, Chow and Jang (2014) added Allee effect to Eq (1) which becomes
\begin{equation}
\left\{\begin{array}{ll}
x(t+1)= \frac{a_1 x(t)}{1+x(t)+c_1 y(t)} \frac{x(t)}{m_1 +x(t) },\[2ex]
y(t+1)= \frac{a_2 y(t)}{1+c_2 x(t)+ y(t)} \frac{y(t)}{m_2 +y(t) }.
\end{array}\right.
\end{equation}
The global behavior of the system, which may possess four interior steady states, is clarified. Study on the special case that $d=3$ and all $ c_{ij}=c$ for $i \not = j$ will be reported. |
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