Codimension two and three bifurcations of a predator–prey system with group defense and prey refuge

A predator–prey system with nonmonotonic functional response and prey refuge is considered. We mainly obtain that the system has the bifurcations of cusp-type codimension two and three, these illustrate that the dynamic behaviors of the model with prey refuge will become more complicated than the system with no refuge

A Predator-Prey Model with Non-Monotonic Response Function

We study the dynamics of a family of planar vector fields that models certain populations of predators and their prey. This model is adapted from the standard Volterra-Lotka system by taking into account group defense, competition between prey and competition between predators. Also we initiate computer-assisted research on time-periodic perturbations, which model seasonal dependence. We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multi-stability occurs. Further, we study the bifurcations between the various domains of structural stability. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. We find several codimension 3 bifurcations that form organizing centers for the global bifurcation set. Studying the time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors

Dynamics of a Leslie-Gower type predator-prey system with herd behavior and constant harvesting in prey

In this paper, the dynamics of a Leslie-Gower type predator-prey system with herd behavior and constant harvesting in prey are investigated. Earlier work has shown that the herd behavior in prey merely induces a supercritical Hopf bifurcation in the classic Leslie-Gower predator-prey system in the absence of harvesting. However, the work in this paper shows that the presence of herd behavior and constant harvesting in prey can give rise to numerous kinds of bifurcation at the non-hyperbolic equilibria in the classic Leslie-Gower predator-prey system such as two saddle-node bifurcations and one Bogdanov-Takens bifurcation of codimension two at the degenerate equilibria and one degenerate Hopf bifurcation of codimension three at the weak focus. Hence, the research results reveal that the herd behavior and constant harvesting in prey have a strong influence on the dynamics and also contribute to promoting the ecological diversity and maintaining the long-term economic benefits.Comment: 20 pages, 10 figure

Multiple wave solutions in a diffusive predator-prey model with strong Allee effect on prey and ratio-dependent functional response

A thorough analysis is performed in a predator-prey reaction-diffusion model which includes three relevant complex dynamical ingredients: (a) a strong Allee effect; (b) ratio-dependent functional responses; and (c) transport attributes given by a diffusion process. As is well-known in the specialized literature, these aspects capture adverse survival conditions for the prey, predation search features and non-homogeneous spatial dynamical distribution of both populations. We look for traveling-wave solutions and provide rigorous results coming from a standard local analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections and periodic orbits in the associated dynamical system in $R^4$. In so doing, we present and describe a diverse zoo of traveling wave solutions; and we relate their occurrence to the Allee effect, the spreading rates and propagation speed. In addition, homoclinic chaos is manifested via both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An actual computation of global invariant manifolds near a focus-focus homoclinic bifurcation is also presented to enravel a multiplicity of wave solutions in the model. A deep understanding of such ecological dynamics is therefore highlighted.Comment: 35 pages, 22 figure

A discrete-time dynamical model of prey and stage-structured predator with juvenile hunting incorporating negative effects of prey refuge

This paper examines a discrete predator-prey model that incorporates prey refuge and its detrimental impact on the growth of the prey population. Age structure is taken into account for predator species. Furthermore, juvenile hunting as well as prey counter-attack are also considered. This paper provides a comprehensive analysis of the existence and stability conditions pertaining to all possible fixed points. The analytical and numerical investigation into the occurrence of different bifurcations, such as the Neimark-Sacker bifurcation and period-doubling bifurcation, in relation to various parameters is discussed. The impact of the parameters reflecting prey growth and prey refuge is thoroughly addressed. Numerous numerical simulations are presented in order to validate the theoretical findings

Dynamic analysis of a Leslie-Gower predator-prey model with the fear effect and nonlinear harvesting

In this paper, we investigate the stability and bifurcation of a Leslie-Gower predator-prey model with a fear effect and nonlinear harvesting. We discuss the existence and stability of equilibria, and show that the unique equilibrium is a cusp of codimension three. Moreover, we show that saddle-node bifurcation and Bogdanov-Takens bifurcation can occur. Also, the system undergoes a degenerate Hopf bifurcation and has two limit cycles (i.e., the inner one is stable and the outer is unstable), which implies the bistable phenomenon. We conclude that the large amount of fear and prey harvesting are detrimental to the survival of the prey and predator

Prey switching with a linear preference trade-off

In ecology, prey switching refers to a predator's adaptive change of habitat or diet in response to prey abundance. In this paper, we study piecewise-smooth models of predator-prey interactions with a linear trade-off in a predator's prey preference. We consider optimally foraging predators and derive a model for a 1 predator-2 prey interaction with a tilted switching manifold between the two sides of discontinuous vector fields. We show that the 1 predator-2 prey system undergoes a novel adding-sliding-like (center to two-part periodic orbit; “C2PO'') bifurcation in which the prey ratio transitions from constant to time-dependent. Farther away from the bifurcation point, the period of the oscillating prey ratio doubles, which suggests a possible cascade to chaos. We compare our model predictions with data on freshwater plankton, and we successfully capture the periodicity in the ratio between the predator's preferred and alternative prey types. Our study suggests that it is useful to investigate prey ratio as a possible indicator of how population dynamics can be influenced by ecosystem diversity

Threshold control strategy for a Filippov model with group defense of pests and a constant-rate release of natural enemies

In this paper, we establish an integrated pest management Filippov model with group defense of pests and a constant rate release of natural enemies. First, the dynamics of the subsystems in the Filippov system are analyzed. Second, the dynamics of the sliding mode system and the types of equilibria of the Filippov system are discussed. Then the complex dynamics of the Filippov system are investigated by using numerical analysis when there is a globally asymptotically stable limit cycle and a globally asymptotically stable equilibrium in two subsystems, respectively. Furthermore, we analyze the existence region of a sliding mode and pseudo equilibrium, as well as the complex dynamics of the Filippov system, such as boundary equilibrium bifurcation, the grazing bifurcation, the buckling bifurcation and the crossing bifurcation. These complex sliding bifurcations reveal that the selection of key parameters can control the population density no more than the economic threshold, so as to prevent the outbreak of pests

Dynamics of a harvested cyanobacteria-fish model with modified Holling type Ⅳ functional response

In this paper, considering the aggregation effect and Allee effect of cyanobacteria populations and the harvesting of both cyanobacteria and fish by human beings, a new cyanobacteria-fish model with two harvesting terms and a modified Holling type Ⅳ functional response function is proposed. The main purpose of this paper is to further elucidate the influence of harvesting terms on the dynamic behavior of a cyanobacteria-fish model. Critical conditions for the existence and stability of several interior equilibria are given. The economic equilibria and the maximum sustainable total yield problem are also studied. The model exhibits several bifurcations, such as transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. It is concluded from a biological perspective that the survival mode of cyanobacteria and fish can be determined by the harvesting terms. Finally, concrete examples of our model are given through numerical simulations to verify and enrich the theoretical results