Narrow Results

In this paper, we are concerned with the propagation dynamics of a nonlocal dispersal predator–prey model with one predator and two preys. Accurately, we mainly study the invading phenomenon of an alien predator to the habitat of two aborigine preys, which is depicted by traveling waves connecting the predator-free state to the co-existence state. We characterize the minimal wave speed of this invading process based on an application of Schauder’s fixed point theorem with the help of generalized upper-lower solutions and the Lyapunov argument. Particularly, the discussion of traveling waves with critical wave speed is more involved due to the effect of nonlocal dispersal. Finally, a numerical simulation is given to present the traveling waves and shows the differences between non-local and local dispersals.

In this paper, a modified Leslie-type predator–prey system with Holling-type III functional response is formulated to investigate the dynamics in the presence of special Allee effect induced by fear factors. First, we calculate the first three focal values by the method of successor function to ensure that the system has an unstable weak focus of multiplicity 3. A focus- or center-type degenerate Bogdanov–Takens singularity of codimension 3 and a weak focus of multiplicity 1 or 2 together with a cusp of codimension 2 are derived for the system. Then multiple bifurcations are explored. It is shown that the system undergoes Hopf bifurcation of codimension 3 and hence three limit cycles are generated. There exists another large stable limit cycle enclosing these three limit cycles by Poincaré–Bendixson theorem. We demonstrate that a degenerate focus-type Bogdanov–Takens bifurcation of codimension 3 can occur by calculating the universal unfolding. It is also shown that a subcritical, supercritical or degenerate Hopf bifurcation occurs together with a Bogdanov–Takens bifurcation. Finally, numerical simulations are used to support the theoretical results. The analysis results reveal that the special Allee effect is beneficial for the persistence and diversity of ecosystem.

Invasive species can inflict significant harm on biodiversity and disrupt ecosystem stability. Introducing pathogens to combat invasive species is recognized as an effective method for preserving biodiversity. Against this backdrop, we examine a predator–prey model involving diseased predators, taking into account the impact of environmental noise, and then propose a stochastic predator–prey model where predators carry diseases. In this model, we assume that the intrinsic growth rate of the prey and the disease transmission rate of the predator follow an Ornstein–Uhlenbeck process. Our contributions in this study can be summarized as follows: First, we establish the existence and uniqueness of the global solution for the stochastic predator–prey model, along with an analysis of moment estimation for the global solution. Second, we provide sufficient conditions for the existence of stationary distribution and population extinction within the stochastic predator–prey model. Furthermore, we derive the precise expression of the probability density function near the quasi-positive equilibrium $E_0^{\ast }$ of the stochastic predator–prey model. Lastly, we validate our analytical findings through numerical simulations.

This piece of work circles around the investigation of a predator–prey model that incorporates nonlinear harvesting and group defense, both of which are important phenomena occurring in the ecosystem. To analyze the proposed model, both analytical and numerical methods are employed. The model experiences a series of bifurcations, including saddle-node, Hopf and Bogdanov–Takens. The system exhibits bifurcations of codimension one and two but no bifurcation of codimension three emerged. Phase portrait diagrams are also presented to confirm the validity of both the analytical and numerical findings.

In this paper, a discrete version of continuous non-autonomous predator–prey model with delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and globally asymptotically stability of positive periodic solution of difference equations in consideration are established. Finally, some numerical examples are given to verify the theoretical analysis.

In this paper, we consider a stochastic nonautonomous predator–prey model with modified Leslie–Gower and Holling II schemes in the presence of environmental forcing. The deterministic model is the modified Holling–Tanner model which is an extension of the classical Leslie–Gower model. We show that there is a unique positive solution to the stochastic system for any positive initial value. Sufficient conditions for strong persistence in mean and extinction to the stochastic system are established.

We presented a predator–prey lattice model containing moveable wolves and sheep, which are characterized by Penna double bit strings. Sexual reproduction and child-care strategies are considered. To implement this model in an efficient way, we build a fully parallel Cellular Automata based on a new definition of the neighborhood. We show the roles played by the initial densities of the populations, the mutation rate and the linear size of the lattice in the evolution of this model.

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

Spatial patterns have influence on modifying the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, we have considered a predator-prey model taking into account both diffusion and migration. We present a theoretical analysis of the condition of emerging spatial pattern. The results of numerical simulations reveal that the migration has marked effect on the pattern formation of the population, i.e. changing Turing pattern to be traveling pattern. The obtained results show that the modeling by both migrations and diffusion can account for the dynamic complexity of ecosystems.

A two species predator–prey model based on the Lotka–Volterra equations is proposed, where the fitness of an individual animal depends upon the relative strength of its genes. Simulations of the model show that the system passes from the standard oscillatory solution of the Lotka–Volterra equations into a steady-state regime, which exhibits many of the characteristics of self-organized criticality, including a 1/f power spectrum.

We subject the classical Volterra predator–prey ecosystem model with age structure for predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of prey. By using the Poincaré surface-of-section technique and numerical experiments, an abundance of periodic solutions, quasiperiodic solutions and chaotic solutions are revealed. The route of transitions to chaos were found to be very complex which include quasiperiodicity and frequency locking.

Based on the classical Lotka–Volterra predator–prey system, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. We observe that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Numerical results show that the system we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and nonunique dynamics, meaning that several attractors coexist. Finally, a pest–predator stage-structured model for the pest concerning this kind of impulsive control strategy is proposed, and we also show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some threshold.

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator–prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.

Bazykin proposed a Lotka–Volterra-type ecological model that accounts for simplified territoriality, which neither depends on territory size nor on food availability. In this study, we describe the global dynamics of the Bazykin model using analytical and numerical methods. We specifically focus on the effects of mutual predator interference and the prey carrying capacity since the variability of each could have especially dramatic ecological repercussions. The model displays a broad array of complex dynamics in space and time; for instance, we find the coexistence of a limit cycle and a steady state, and bistability of steady states. We also characterize super- and subcritical Poincaré–Andronov–Hopf bifurcations and a Bogdanov–Takens bifurcation. To illustrate the system's ability to naturally shift from stable to unstable dynamics, we construct bursting solutions, which depend on the slow dynamics of the carrying capacity. We also consider the stabilizing effect of the intraspecies interaction parameter, without which the system only shows either a stable steady state or oscillatory solutions with large amplitudes. We argue that this large amplitude behavior is the source of chaotic behavior reported in systems that use the MacArthur–Rosenzweig model to describe food-chain dynamics. Finally, we find the sufficient conditions in parameter space for Turing patterns and obtain the so-called "back-eye" pattern and localized structures.

In this work, we consider a three-dimensional predator-prey model with impulsive harvesting and diffusion at different fixed moments. We prove that all solutions of the investigated system are uniformly ultimately bounded. The conditions of the globally asymptotically stable prey-extinction boundary periodic solution of the investigated system are obtained, as well the permanence of the investigated system. Finally, numerical analysis is inserted to illustrate the results which provide reliable tactic basis for the practical biological resource management.

In this paper, a model of predator-prey with disease in food chain is investigated — where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis.

A predator–prey system with neutral delay is investigated from the viewpoint of bifurcation analysis on neutral delay differential equations. Stability and Hopf bifurcation of the inner equilibrium are given, by which we show how the neutral terms affect the dynamical behavior of the prey and the predator. To give more detailed information on the periodic oscillations, the direction and stability of Hopf bifurcation are studied by using the normal form theory of neutral equation. We find neutral delay makes the predator–prey system more complicated and usually induces stability switches or double Hopf bifurcations. Near the double Hopf bifurcation we give the detailed bifurcation set by calculating the universal unfoldings. It is shown that the population of prey or predator may exhibit transient quasiperiodic oscillations driven by the neutral delay. Finally, we carry out several groups of illustrations.

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

In the paper, we study the existence of periodic solutions in an invariant manifold for a predator–prey model with periodic harvesting rate. Without harvesting, the predator–prey system has a positive equilibrium of center type, which undergoes Hopf bifurcation. Under the periodic harvesting, we first reduce this predator–prey system to a complex normal form. Then, we define the Poincaré mapping of this system by using the polar coordinates, and prove that there exists an integral manifold and periodic solution with the period of the harvesting rate. An example is provided to illustrate the result.

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie–Gower functional response as well as that of Beddington–DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values.