Explosive tritrophic food chain models with interference: A comparative study

Highlights

• The effect of top predator interference on the dynamics of a tritrophic food chain model is the principal goal to study general framework for three species food chain models in which intermediate and top predators are specialist and generalist types respectively where the top predator grows by sexual reproduction.

• This concept is used to design five different models in this paper by five different combinations of functional responses of specialist intermediate predator (food uptake process follows either prey dependent, Holling type III/IV or prey-predator dependent, BD-functional response) and sexually reproductive generalist top predator (food uptake follows strictly prey-predator dependent, BD/CM functional response) respectively assuming the fact that the growth of intermediate predator is mediated by gestation delay.

• Stability of each equilibria of the non-delayed counterpart of each model and the existence of Hopf-bifurcation for the coexisting equilibrium point of all delayed systems are established.

• The delayed and non-delayed models can blow-up in finite time under sufficient conditions on the initial data. Furthermore, some dynamic behaviors of the systems are clarified by some numerical tests. The numerical simulations show that the gestation delay can act as a damping mechanism and prevent blow-up in certain critical range of gestation period.

• Ecological implications of this phenomenon are discussed.

Abstract

Depending upon the choice of food, availability of resource and growth structure, food uptake process of higher trophic level species are significantly complicated and gives interesting dynamical impacts on community food chain. The effect of top predator interference on the dynamics of a tritrophic food chain model is the principal goal to study general framework for three species food chain models in which intermediate and top predators are specialist and generalist types respectively where the top predator grows by sexual reproduction. This concept is used to design five different models in this paper by five different combinations of functional responses of specialist intermediate predator (food uptake process follows either prey dependent, Holling type III/IV or prey-predator dependent, Beddington-DeAngelis (BD)-functional response) and sexually reproductive generalist top predator (food uptake follows strictly prey-predator dependent, BD/Crowley–Martin (CM) functional response) respectively assuming the fact that the growth of intermediate predator is mediated by gestation delay. We establish the stability of each equilibrium point of the non-delayed counterpart of each model and investigate the existence of Hopf-bifurcation for the coexistence equilibrium point of all delayed systems. We also show that both the delayed and non delayed models can blow-up in finite time under sufficient conditions on the initial data. Furthermore, some dynamic behaviors of the systems are clarified by some numerical tests. The numerical simulations show that the gestation delay can act as a damping mechanism and prevent blow-up in certain critical range of gestation period. Ecological implications of this phenomenon are discussed.

Introduction

Ecological models are increasingly used to emphasize the linkage and relationship between social and physical environments. They demonstrate our responses to such factors over entire life cycles. They are easy to understand and interpret, they are often based on causality and they can include time delay, age/stage structure, functional response, etc. The effect of these factors has illustrated complicated and rich dynamics such as stability, bifurcation, periodic solutions, persistence, etc [1]. Research results herein are helpful to predict the developing/decaying tendency of specific populations to determine the key factors behind extinction and to seek optimum strategies of preventing and controlling extinction of various species [2].

In population dynamics, a functional response refers to the change in the density of prey consumed per unit time by per predator, as the prey density changes [3]. Based on different backgrounds and experimental data, various forms of functional responses have been developed mimicing various processes of energy transfer in predator-prey systems [4]. Categories which specify the functional response: prey dependent (function of only prey density, viz. Holling type I - IV [5]) and prey-predator dependent (function of both prey-predator density, viz. Beddington-DeAngelis [6], Crowley-Martin [7], Hassel-Varley [8] etc) those are describing the predator’s interference on their hunting procedure. The inherent feature of these functions means that the more prey in the environment the better off the predator, which is true in many predator-prey interactions [9]. Also, non-monotonic response occurs at the microbial level when the nutrient concentration reaches a high level an inhibitory effect on the specific growth rate may occur, Holling type IV functional response are used to model such inhibitory effect. Some researchers suggest that a more suitable general predator-prey model should have a prey-predator dependent functional response [10], [11], [12], [13] and all the prey-predator dependent functional responses can provide better description of predator feeding over a range of predator-prey abundances present [14]. Feeding procedure of higher trophic level species is more complicated than its below trophic level species [15], [16]. In general it is shown in the nature that if lower trophic level species is specialist type, then there is high chance to be generalist type of the higher trophic level species [15], [16]. Sexual reproduction is very common process for the growth of that kind of higher trophic level generalist species [17], [18], [19], [20], [21]. Also gestation period is very important factor in population dynamics and assuming that reproduction of predator after consuming prey is not instantaneous but mediated by some time lag required for gestation [22], [23]. It has been shown that many models in the class of where the top predator is modeled via the modified Leslie-Gower scheme for sexually reproductive generalist predator [22], [24].

We first provide some ecological background of invasive species. We are motivated primarily to control non-native species, which is a cental problem in spatial ecology. Data on species such as the invasive Burmese python (Python bivittatus) in the Florida everglades, show an exponential increase in python population, which have resulted in local prey population reducing severely [25]. An invasive species is formally defined as any species capable of propagating itself in a nonnative environment and thus establish a self-sustained population. In the United States alone damages caused by invasive species to agriculture, forests, fisheries and businesses, have been estimated to be $120 billion a year [26]. Another potential issue is that the environment may turn favorable for a certain species while becoming unfavorable for its competitors or natural enemies. This can result in the favored species to outbreak [27]. For example, in the European Alps certain seasonal environmental conditions enable the population of the larch budmoth to become large enough to defoliate entire forests [28]. then for sufficiently large initial data of top predator density, a potential to blow-up/explode in finite time can occur [24]. Biological control is an adopted strategy to limit harmful populations [29]. The objective of a biological control is to establish a management strategy that best controls and decreases the harmful population to healthy levels as opposed to high and risky levels. Naturally, how does one define high level, and further, how well does the biological control actually work, at various high levels? We have recently started investigating this question via the mathematical property of finite time blow-up [30], [31], [32].

This ideas have also been applied to model invasive populations that seems to be “exploding”, under a variety of ecological scenarios. In the current manuscript we ask:

• What is the foraging pressure of intermediate specialist predator on prey if it is modeled by Lotka–Volterra scheme and generalist sexually reproductive top predator on intermediate predator if the top predator is modeled according to the modified Leslie–Gower scheme?

• What are the dynamical outcomes when the predation process of intermediate and top predator are considered by different combinations of prey dependent and prey-predator dependent functional responses? With respect to the different combinations of functional responses different models will come for analysis.

• We ask what would the effects be, if there is predator interference as well.

• What will be periodic effect of gestation delay of intermediate predator on the steady state of the system?

• In particular, we ask what would the finite time blow up be, if the top generalist predator grows by sexual reproduction as well.

• How would the blow-up/blow up-prevention be effected in the case of various functional responses/feeding rates of the preys.

In this study, the effect of top predator interference on the dynamics of a general tri-trophic food chain model with five different combinations of functional responses is investigated. The manuscript is organized as follows: Models are formulated in Section 2 and stability properties of these models are studied in Section 3. Also, existence of Hopf-bifurcation for coexisting equilibrium point of all models are investigated in this section. In Section 4, finite time blow up behavior is investigated and numerical simulation has been carried out in Section 5. A comparative study between different models and a brief conclusion are in Sections 6 and 7 respectively.