Comparison theorem for systems of differential equations and its application to estimate the average time benefit from resource collection

Abstract

One of the versions of the comparison theorem for systems of ordinary differential equations is proved, the consequence of which is the property of monotonicity of solutions with respect to initial data. We consider the problem of estimating the average time profit from resource extraction for a structured population consisting of individual species $x_1,\ldots,x_n$, or divided into $n$ age groups. We assume that the dynamics of the population in the absence of exploitation is given by a system of differential equations $\dot x =f(x)$, and at times $\tau(k)=kd$, $d>0$, a certain share of the biological resource is extracted from the population $u(k)=(u_1(k),\ldots,u_n(k))\in [0,1]^n$, $k=1,2,\ldots.$ It is shown that using the comparison theorem it is possible to find estimates of the average time benefit in cases when analytical solutions for relevant systems are not known. The results obtained are illustrated for models of interaction between two species, such as symbiosis and competition. It is shown that for models of symbiosis, commensalism and neutralism, the greatest value of the average time profit is achieved with the simultaneous exploitation of two types of resources. For populations between which an interaction of the “competition” type is observed, cases in which it is advisable to extract only one type of resource are highlighted.