Sources, sinks and selectivity (Q2745332)

The authors develop a model which allows for multiple habitat patches with a range of habitat qualities among the various patches. Also they provide for the possibility that the patch selection priority may not follow the ranking of expected reproductive success. The model consists of several habitat patches which vary with regard to size, reproduction and mortality rates of residents of the patch, and attractiveness to those animals which are searching for a patch to occupy (settling rate). They tracked only the reproducing animals, the females, presuming that finding a mate is not a problem. It is assumed that once established in a particular partch, the resident remains there and reproduces until death, at which time her space becomes available to another searching animal. Meanwhile, new offspring are absorbed by the population of searchers. If \(K\) is the number of patches, the model is an autonomous dissipative monotone system of \(K+1\) first order nonlinear homogeneous ordinary differential equations, of second degree in the simplest instance. These equations couple the population of searchers with the residents of each of the patches, but there is no interaction among residents. This system is a logical extension of a two-dimensional system which has been analyzed previously. Exploring it, in this paper they search for equilibria, their stability, and the patterns of approach to the stable equilibria. The authors are particulary interested in conditions that prove for multiple stable equilibria and the consequences of such a situation. Throughout this paper, lowercase Latin and Greek letters will denote constants or functions of \(s\), upper case Latin letters will be constants or functions of \(t\), and it will use `prime'' notation for all derivatives rather than \(d/dt\) or \(/ds\). The model is described by the following system of differential equations: NEWLINE\[NEWLINEN^{'}_{i}= \phi_{i}(S,N)=a_{i}(S)S(h_{i}-N_{i})-m_{i}N_{i} (1)NEWLINE\]NEWLINE NEWLINE\[NEWLINE = \alpha _{i}(S)(h_{i}-N_{i})-m_{i}N_{i} (2),\quad \text{for}\quad i=1\dots,K,NEWLINE\]NEWLINE NEWLINE\[NEWLINES^{'}=\phi(S,N)=\sum^{K}_{i=1}(b_{i}(S)N_{i}-\alpha_{i}(S)(h_{i}-N_{i}))-m(S)S.NEWLINE\]NEWLINE The authors have used the system of equations \((1)\) and \((2)\) to model a territorial population for which individuals have several choices of habitat and /or may not make the best choices considering the birth and death and rates associated with each patch. This system turns out to be surprisingly tractable. The fixed points and, almost always, their stability status, can be read directly off the graph of the function \(g\), which in the constant case is shaped roughly like that of a polynomial, thanks to the fact that the system is cooperative and there is no interaction among the \(N_{i}.\) Monotone flow also describes the eventual direction of trajectories as they were initially visualized through simulations. Finally, one has been able to get partial information about basins of attraction.