A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters

doi:10.1016/j.amc.2010.05.049

Applied Mathematics and Computation

Volume 217, Issue 3, 1 October 2010, Pages 1213-1219

https://doi.org/10.1016/j.amc.2010.05.049 Get rights and content

Abstract

In this work, we propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, which goes beyond the existing theory, and for the special case of period-two parameters we analytically show the conjecture is true. For this case we show that the stability region in parameter space obtained from the conjecture is larger than a previously proposed stability region. The period-three case is investigated numerically and similar extensions are realized. This suggests that the current theory cited in this paper, while giving sufficient conditions for stability is far from optimal.

Introduction

Recent advances in genetics have allowed scientists to genetically modify mosquitoes in the laboratory to hinder or block parasite transmission, thus making the mosquitoes refractory. This opens the possibility to release genetically modified mosquitoes into the wild with the objective to reduce the spread of mosquito borne diseases like Malaria. The progress in this area is fairly recent [1] and only few mathematical models for the population dynamics of wild and genetically modified mosquitoes are available in the literature. Li [2] proposed a discrete-time mathematical model for populations consisting of wild and genetically altered mosquitoes, and in [3] presented an extension of his model where the zygocity of the mosquitos is considered. See Ackleh et al. [4] for a three-stage discrete-time population model with continuous versus seasonal reproduction, and Jang [5] for a discrete-time model consisting of two interacting populations. Ratio dynamics was used in [7], [8] to decouple the mosquitoes population dynamics equations introduced in [2] into two Ricker equations. In [6], the idea of dynamic reduction was introduced, and a model where the seasonal variability was taken into account by allowing the birth and survival functions to have periodic parameters was presented. Through the use of dynamic reduction the equations governing the population of the wild and genetically modified mosquitoes become a set of decoupled Ricker equations with periodic parameters. So the study of the stability of the periodic solutions of the mosquitoes model in [6] reduces to study the stability of periodic solutions of the Ricker equation with periodic parameters. This highlights the importance of understanding the stability of the periodic solutions of Ricker’s equation with periodic parameters and the motivation for the work here presented.

Here we propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, and for the special case of period-two parameters we analytically show the conjecture is valid. Furthermore, for this case we show that the stability region in parameter space obtained from the conjecture is larger than the one obtained by applying the results of Zhou and Zou [9]. Results from numerical solutions suggest that for some regions in parameter space, it may be possible to attain stability of the periodic solutions for parameter values beyond the ones constrained by the conjecture.

The conjecture on the stability of periodic solutions of Ricker’s equation with periodic parameters is given in Section 2, where it is also shown that the conjecture is valid for periodic parameters with period-two. In Section 3 the conjecture is explored numerically for parameters with periods 2, 3 and 4. Conclusions are given in Section 4.

Section snippets

The conjecture

In this section we present a conjecture on the stability of the periodic orbit for the Ricker equation with periodic parameters. The stability of such equations has been studied by Zhou and Zou [9], Kon [10], and Sacker [13].

Consider the Ricker equation given by $z_{n + 1} = z_{n} e^{p (n) - z_{n}},$ where the parameter p(n) is periodic with period k. In the trivial case of k = 1 we have that p(n) = p for all n, and (2.1) has the unique positive fixed point z∗ = p, that is globally asymptotically stable with respect to

Results from numerical simulations

Numerical simulations were conducted in order to estimate the values of ϵ_k for k = 2, 3, 4. For each k a large set of randomly selected parameters p(0), p(1), … , p(k − 1) satisfying the conditions of the conjecture was selected. At every combination of the parameters the fixed point of (2.1) was computed together with the derivative at the fixed point. The largest possible values of ϵ_k for which all parameter combinations yielded a stable fixed point were selected. The numerical simulations gave as

Conclusions

A conjecture about the stability of periodic solutions of the Ricker equation with periodic parameters is given. For the special case of period-two parameters the conjecture is shown to be valid. The region in parameter space (for period-two parameters) given by the conjecture where the periodic solutions of the equation are asymptotically stable is compared with the region obtained by a result from Zhou and Zou [9]. The stability region obtained by using the conjecture encompasses the one

References (13)

Jia Li
Simple mathematical models for mosquito populations with genetically altered mosquitos
Math. Biosci.
(2004)
Zhan Zhou et al.
Stable periodic solutions in a discrete logistic equation
Appl. Math. Lett.
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Flaminia Catteruccia et al.
Stable germline transformation of the malaria mosquito Anopheles stephensi
Nature
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Jia Li
Heterogeniety in modelling of mosquito populations with transgenic mosquitoes
J. Difference Eq. Appl.
(2005)
Azmy S. Ackleh et al.
A three-stage discrete-time population model: continuous versus seasonal reproduction
J. Biol. Dyn.
(2007)
Sophia R.-J. Jang
On a discrete West Nile epidemic model
Comput. Appl. Math.
(2007)

There are more references available in the full text version of this article.

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¹: Supported by University of Southern California, Letters Arts and Sciences Faculty Development Grant.

²: Supported by The California State Polytechnic University, Research, Scholarship and Creative Activity Grant.

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Applied Mathematics and Computation

Abstract

Introduction

Section snippets

The conjecture

Results from numerical simulations

Conclusions

References (13)

Simple mathematical models for mosquito populations with genetically altered mosquitos

Math. Biosci.

Stable periodic solutions in a discrete logistic equation

Appl. Math. Lett.

Stable germline transformation of the malaria mosquito Anopheles stephensi

Nature

Heterogeniety in modelling of mosquito populations with transgenic mosquitoes

J. Difference Eq. Appl.

A three-stage discrete-time population model: continuous versus seasonal reproduction

J. Biol. Dyn.

On a discrete West Nile epidemic model

Comput. Appl. Math.

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On the proper periodic Ricker model

Discrete Melnikov functions