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

The SIS-Model on Time Scales

In this paper, we introduce the epidemic model following the hypothesis of the disease flow Susceptible → Infected → Susceptible, short SIS, on time scales. After a brief introduction of time scales, we present dynamic systems representing the SIS-model on time scales and derive its solution sets. We are discussing the stability of the steady states before investigating a modified SIS-model including a birth and death rate. Throughout, examples are used to illustrate the results. 2010 Mathematics Subject Classification: 34N05, 92D25

Abel dynamic equations of first and second kind

This paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined

The Second Cushing-Henson Conjecture for the Beverton-Holt q-Difference Equation

In this paper, we study the second Cushing-Henson conjecture for the Beverton-Holt difference equation with periodic inherent growth rate and periodic carrying capacity in the quantum calculus setting. We give a short summary of recent results regarding the Beverton-Holt difference and q-difference equation and introduce the theory of quantum calculus briefly. Next, we analyze the second Cushing-Henson conjecture. We extend recent studies in [The Beverton-Holt q-difference equation with periodic growth rate, Difference Equations, Discrete Dynamical Systems, and Applications, Springer-Verlag, Berlin, Heidelberg, New York, 2015, pp. 3-14] and state a modified formulation of the second Cushing-Henson conjecture for the Beverton-Holt q-difference equation as a generalization of existing formulations

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