On an Extended Time-Varying Beverton–Holt Equation Subject to Harvesting Monitoring and Population Excess Penalty

Abstract

This paper considers a more general eventually time-varying Beverton–Holt equation for species evolution which can include a harvesting action and a penalty for overpopulation numbers. The harvesting action may be positive (typically consisting of hunting or fishing) or negative which refers to repopulation within the environment. One considers also a penalty of quadratic type on the overpopulation and the introduction of a term related to Allee effect to take account of small levels of population. The intrinsic growth rate is assumed either to exceed unity or to be under unity. In the second case, the extinction point is a locally stable attractor while the other positive equilibrium point is unstable contrarily to the commonly studied case of intrinsic growth rate exceeding unity where the above roles are inverted. This consequence implies that the extinction point is also globally asymptotically stable for any given finite initial condition. In the case when the eventual overpopulation is penalized with a sufficiently large coefficient which exceeds a prescribed threshold, to quantify such an excess, only a globally asymptotically stable extinction attractor is present and no other positive equilibrium points exist. In the case of a positive moderate quadratic evaluation term for such an overpopulation, one or two positive equilibrium points coexist with the extinction one. The smaller one is unstable contrarily to the extinction equilibrium which is locally asymptotically stable. If it exists a second largest positive equilibrium point, being distinct to the above-given one, then it can be unstable or locally stable depending on the parameterization. Also, some methods of monitoring the population evolution through control laws on the harvesting action are discussed.