Derivation and dynamics of discrete population models with distributed delay in reproduction (Q6632654)

The authors extend the discrete delay single-species population model,\N\[\NX_{t+1}=p(X_t)X_t+\tilde{p}(r,X_{t-r})g(X_{t-r})X_{t-r},\N\]\Nto the following discrete distributed delay model,\N\[\NX_{t+1}=p(X_t)X_t+\sum\limits_{s=\tau}^{\tau+\tau_M}K(s) \tilde{p}(s,X_{t-s})g(X_{t-s})X_{t-s}, \qquad t\in \mathbb{N}_0=\{0,1,2,\ldots\}.\N\]\NHere \(X_t\) is the mature subpopulation of the species at time \(t\). Under reasonable assumptions based on biology, the solution of the model is either trivial or positive for \(t\ge \tau+\tau_M\) and bounded. First, for the general case with the function \(g\), it is shown that there is a critical delay threshold \(\tilde{\tau}_c\) (which is defined as the smallest possible nonnegative integer such that the trivial equilibrium is globally asymptotically stable for all non-negative initial data, whenever \(\tau\ge \tilde{\tau}_c\)). Then, the special case \(g(x)=\bar{g}>0\) is studied. Not only an equation determining \(\tilde{\tau}_c\) for fixed \(\tau_M\) but also a lower bound on \(\tau_M\) for fixed \(\tau\), \(\tilde{\tau}_M\) (which guarantees the extinction if \(\tau_M\ge \tilde{\tau}_M\)), are provided. Moreover, a comparison of the critical delay thresholds for different kernels \(K(s)\) indicates that to avoid extinction it is best if all individuals in the population have the shortest delay possible. Finally the modeling idea is applied to a Beverton-Holt model and the global dynamics is investigated. In particular, for kernels sharing the same mean delay, populations with the largest variance in the time required to reach maturity have the lowest chance of extinction.