Candace M. Kent

Global behavior of solutions of a periodically forced Sigmoid Beverton–Holt model

Our aim in this paper is to investigate the boundedness, the extreme stability, and the periodicity of positive solutions of the periodically forced Sigmoid Beverton–Holt model: where {a n } is a positive periodic sequence with period p and δ>0. In the special case when δ=1, the above equation reduces to the well-known periodic Pielou logistic equation which is known to be equivalent to the periodically forced Beverton–Holt model.

On the Difference Equation

We show that the difference equation , where , the parameters , and initial values , are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.

On the boundedness of positive solutions of the reciprocal max-type difference equation xn = max { A1n-1 / xn-1, A2n-1/xn-2, …, Atn-1/xn-t } with periodic parameters

We investigate the boundedness of positive solutions of the reciprocal max-type difference equation xn = max { A1n-1 / xn-1, A2n-1/xn-2, …, Atn-1/xn-t } where, for each value of i, the sequence { Ain}∞n=0 of positive numbers is periodic with period pi. We give both sufficient conditions on the pis for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded. This work essentially complements the work by Bidwell and Franke, who showed that as long as every positive solution of our equation is bounded, then every positive solution is eventually periodic, thereby leaving open the question as to when solutions are bounded.

Solutions of the Difference Equation

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: 𝑥𝑛

Management of invasive Allee species

Abstract In this study, we use a discrete, two-patch population model of an Allee species to examine different methods in managing invasions. We first analytically examine the model to show the presence of the strong Allee effect, and then we numerically explore the model to test the effectiveness of different management strategies. As expected invasion is facilitated by lower Allee thresholds, greater carrying capacities and greater proportions of dispersers. These effects are interacting, however, and moderated by population growth rate. Using the gypsy moth as an example species, we demonstrate that the effectiveness of different invasion management strategies is context-dependent, combining complementary methods may be preferable, and the preferred strategy may differ geographically. Specifically, we find methods for restricting movement to be more effective in areas of contiguous habitat and high Allee thresholds, where methods involving mating disruptions and raising Allee thresholds are more effective in areas of high habitat fragmentation.

A Proposal for an Application of a Max-Type Difference Equation to Epilepsy

We propose, for the sake of dialogue, that the nonautomomous reciprocal max-type difference equation,

Long-Term Behavior of Solutions of the Difference Equation

\begin{aligned} x_{n+1}=\max \left\{ \frac{A_{n}^{(0)}}{x_{n}}, \frac{A_{n}^{(1)}}{x_{n-1}}, \ldots , \frac{A_{n}^{(k)}}{x_{n-k}}\right\} , \ \ n=0, 1, \ldots , \end{aligned}

Solutions of the Difference Equation xn+1=xnxn-1-1

where the parameters are positive periodic sequences and the initial conditions are positive, when \(k=1\) may serve as a phenomenological model of seizure activity as occurs in mesial (or middle) temporal lobe epilepsy.