Species coexistence and dynamical complexity in three species logistic food webs

The logistic trophic interaction model is built upon a different philosophy than classical Lotka-Volterra models. The logistic model describes the species occupying different trophic levels with the same form of equations while the Lotka-Volterra model describes prey and predator species differently. As such, the logistic trophic interaction model can serve as a framework for a unified theory of trophic interactions. As part of an effort to study the dynamical complexity of logistic food webs, I explore the general dynamic properties of three discrete-time logistic model food webs involving the interaction between three species.; Analysis of a three species logistic food chain indicates that changes to the basal species causes proportional bottom-up effect of the same sign, while changes to the middle species cause top-down effects of the opposite sign and bottom-up effects of the same sign, which then reflect back to, and cause quadratic effects on, the middle species.; Analysis of food webs involving two predator and one prey shows that a second predator may have positive indirect effects on the prey population, and thus mediate the coexistence of an originally non-persistent predator-prey pair, through inter-specific competition with the first predator. Changes in factors affecting one predator may have quadratic effects on the other predator.; Analysis of interaction between one predator and two prey shows many possible outcomes, including: predator mediated coexistence of two competing prey species, coexistence of an originally non-persistent predator-prey pair maintained by the second prey, the possibility of two stable equilibria, and the possibility of two attractors at one equilibrium point.; Discrete-time logistic food webs exhibit a rich array of complex dynamics, including stable equilibria and periodic, quasi-periodic, and chaotic motion. Complex dynamics are dominated by quasi-periodic and chaotic oscillations generated by Hopf-Andronov-Poincare bifurcations. Less common are periodic solutions dominated by period-three or period-five oscillations within biologically reasonable parameter ranges.