Population dynamics of discrete-time predator-prey exploited fishery model

In this dissertation, we use a generalized version of the aggregate production model of Basson and Fogarty in [3] to study the role that both compensatory (non-oscillatory) and overcompensatory (oscillatory) dynamics play in the long term dynamics of exploited fisheries. When each species is governed by compensatory dynamics via the Beverton-Holt model and the predator's response to species interaction is modeled using a linear function, we show that the predator-prey model exhibits a globally stable positive fixed point. In stark contrast, we show that when each species is governed by compensatory dynamics via the Beverton-Holt model and the predator response function is exponential, then the predator-prey model exhibits population oscillations. We use the Ricker model to explore the impact of overcompensatory dynamics on the aggregate production model of Basson and Fogarty. In addition, we investigated the bifurcations of the positive fixed point. In particular, we give specific parameter values for when the predator-prey system undergoes flip bifurcation (stability shifts from the interior fixed point to a 2-cycle) and Neimark-Sacker bifurcation (stability shift from the interior fixed point to an invariant loop). We also illustrate that Basson and Fogarty's predator-prey model exhibits alternate life-history outcomes (multiple attractors).;In [21], Franke and Yakubu showed that a competitive mixed compensatory-overcompensatory system exhibits species coexistence in the absence of a positive equilibrium population. We illustrate a similar result using a mixed compensatory-overcompensatory predator-prey system. That is, we show that when the prey population is governed by compensatory dynamics and the predator population is governed by overcompensatory dynamics, then the predator-prey system supports coexistence in the absence of a positive fixed point. However, our numerical explorations seem to illustrate that the corresponding mixed overcompensatory-compensatory predator-prey systems do not exhibit similar dynamical behaviors.