Derivation and Analysis of a Discrete Predator–Prey Model

We derive a discrete predator–prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark–Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark–Sacker bifurcation as a function of the model parameters

Moving forward in circles: challenges and opportunities in modelling population cycles

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer–resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research

Bifurcation Analysis of a Chemostat Model with a Distributed Delay

A chemostat model of a single species feeding on a limiting nutrient supplied at a constant rate is proposed. The model incorporates a general nutrient uptake function and a distributed delay. The delay indicates that the growth of the species depends on the past concentration of nutrient. Using the average time delay as a bifurcation parameter, it is proven that the model undergoes a sequence of Hopf bifurcations. Stability criteria for the bifurcating periodic solutions are derived. It is also found that the periodic solutions become unstable if the dilution rate is increased. Computer simulations illustrate the results

Chaotic dynamics in a simple predator-prey model with discrete delay

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this compli- cated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, espe- cially since temperature is known to have an effect on the length of certain delays

n-Species Competition in a Periodic Chemostat

A threshold result on the global dynamics of the scalar asymptotically periodic Kolmogorov equation is proved and then applied to models of single species growth and n-species competition in a periodically operated chemostat. The operating parameters and the species specific response functions can be periodic functions of time. Species specific removal rates are also permitted. Sufficient conditions ensure uniform persistence of all of the species and guarantee that the full system admits at least one positive, periodic solution. In the special case when the species specific removal rates are all equal to the dilution rate, the single species growth model, has a threshold between global extinction and uniform persistence, in the form of a positive, periodic coexistence state. Improved results in the case of 3-species competition are also given, including an example illustrating competition mediated coexistence of three species. Short Title: Competition in a Periodic Chemostat Key word..

Derivation and Analysis of a Discrete Predator-Prey Model

We derive a discrete predator-prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark-Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark-Sacker bifurcation as a function of the model parameters