Ivlev-type Functional Response Research Articles

Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

<p>Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.</p>

Existence of wave trains for a diffusive age-structured predator-prey model with Ivlev-type functional response

Existence of wave trains for a diffusive age-structured predator-prey model with Ivlev-type functional response

Pattern dynamics of a predator-prey system with Ivlev-type functional response

Pattern dynamics of a predator-prey system with Ivlev-type functional response

Canard phenomena for a slow-fast predator-prey system with group defense of the prey

In this study, we consider that the predator reproduces much slower than the prey and propose a slow-fast predator-prey system with the group defense of the prey, which is described by the Ivlev-type functional response. We analyze the canard phenomena (relaxation oscillation and canard cycles) of our proposed model by applying the singular perturbation theory. Firstly, we apply the blow-up techniques and visualize the behavior near the special fold point, which allows us to prove the existence of the unique attractive limit cycle and imply the appearance of the canard phenomenon when crossing the fold point, and the numerical simulations verify the theoretical results. Then we further use the singular perturbation theory to investigate the existence and cyclicity of the canard cycle without head and the existence of the canard cycle with head. Our main results of the complex canard phenomenon reveal the fact that some certain species in the ecosystem suddenly burst back many years after being about extinct.

Qualitative analysis of exponential power rate fishery model and complex dynamics guided by a discontinuous weighted fishing strategy

A global data set for 2260 communities of large mammals, invertebrates, plants, and plankton indicated that the predator and prey densities follow a general scaling law (Hotton et al., 2015). Motivated by this study, a predator–prey model for fisheries with an exponential power rate and Ivlev-type functional response is proposed in this work. The influence of exponential factor on the stability of the coexistence equilibrium is investigated. In addition, from the perspective of scientific and reasonable exploitation and utilization of fishery resources, a weighted fishing strategy is introduced into the model, the complex dynamics induced by the weighted harvesting activity is analyzed, which includes the existence of predator-extinction periodic solution, positive order-1 periodic solution, and their stability. The existence threshold condition provides a basis in the process for determining fishing period or frequency, and the stability threshold condition guarantees the robustness of the adapted weighted fishing strategy. In addition, an optimization problem is formulated to ensure a maximum of the revenue during the fishing process by introducing the cost of fishing and sales within one fishing cycle. At the end of the study, numerical simulations and optimization are implemented in MATLAB program to verify the theoretical results and a conclusion is presented from the perspective of biological significance.

Complex dynamical behavior and control of a discrete ecological model

Cyclic types of ecological dynamics have been found in several biological species of forest Lepidoptera. There are several possible reasons for this, for example, interactions with consumers, predators, plant quality index, and interactions of density-dependent type. Furthermore, interaction with consumers and plant quality index is regarded as key ingredients to alter the dynamics for the parasitoid population. Consequently, the quality of food resources fluctuates due to the level of herbs. Such changes have been observed in different systems of forest pests. Lepidoptera (larch budmoth) is a destructive worm that affects high-altitude trees around the world and is rapidly declining and becoming extinct in large areas of the forest. Considering the interaction between the budmoth and plant quality index for larch trees in the mountain range in Switzerland (Swiss Alps), we discuss the dynamics of a discrete-time system. Ivlev type functional response regarding plant quality index is used for the formulation of discrete-time model concerning the interaction between the index of plant quality and budmoth. Moreover, the existence of steady-states, their local behaviors, and the boundedness of solutions are carried out for the discrete-time model under consideration. It is investigated that the model undergoes flip bifurcation about coexistence by applying the theory of normal forms and the center manifold theorem. Furthermore, the direction and the existence of Hopf bifurcation are explored for the model around its coexistence. Various methods of controlling chaos have been introduced to avoid system fluctuations and bifurcating attitudes. Validation of the analytical findings is illustrated through numerical simulations. Finally, the analytical results are validated by experimental and actual field data.

Study of Fear Effect on Prey–Predator Model with Ivlev-Type Functional Response in Fuzzy Environment

A prey–predator model with Ivlev-type functional response and the fear effect on prey species by the predator have been considered for the first time in a crisp as well as fuzzy environment. The effects of fear have been investigated on the stability of the system. Granular function derivative concept has been used to do fuzzy mathematics. For the first time, proper model analysis, positivity, bounds and uniform persistence are studied for our proposed model in fuzzy environment. The conditions of stability of all co-existing equilibrium points and Hopf bifurcation analysis have also been studied in fuzzy environment. Analytical results have been justified by numerical simulation with proper table and graphical presentation in crisp and fuzzy environment both.

Protection zone in a diffusive predator-prey model with Ivlev-type functional response

Protection zone in a diffusive predator-prey model with Ivlev-type functional response

Global Stability of a Predator-Prey Model with Ivlev-Type Functional Response

A predator-prey model with Ivlev-Type functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional responses.

Stability and Hopf bifurcation for a delayed predator–prey model with stage structure for prey and Ivlev-type functional response

In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are made as follows: (1) there are a single predator and a single prey population in the model; (2) the prey is divided by the age into two stage structures, immature and mature; (3) the trophic transfer from prey to predator is incomplete; (4) there are two time delays due to the time of maturation of prey and the time of gestation of predator. Some properties of equilibria for the model without delays are provided. Further, the stability of equilibria and existence of Hopf bifurcation are studied by discussing the different cases of time delays for the model with delays. We observe that delays can cause a stable equilibrium to become unstable one, even occur Hopf bifurcation when delays pass through their corresponding critical values. Meanwhile, we derive explicit formulae to determine the properties of Hopf bifurcation such as the direction of Hopf bifurcation and the stability of periodic solutions. Numerical simulations of all theoretical analyses are given for verifying our theoretical results. In the present work, we demonstrate that the validity and universality of oscillations induced by the time delays both theoretically and numerically. These results of this paper may be helpful for us to further understand the role of the critical values of time delays in stabilizing the predator–prey model.

A study on a Ivlev-type Chemostat model with one competitor producing a toxin

In order to illustrate the phenomena microorganisms produce toxins to compete with its competitors, a Chemostat model of two competitors for single nutrient with Ivlev-type functional response is investigated. The existence of its solutions is analysed, and the local asymptotic stability of the model’s equilibria is obtained. By use of the theory of limit systems and Lyapunov-LaSalle invariant principle, it is shown that its equilibria are globally asymptotically stable under some conditions. Our results demonstrate that the principle of competition exclusion is established, and under different conditions, different microorganism can beat the competition.

Dynamic properties of a delayed predator–prey system with Ivlev-type functional response

In this paper, the dynamical behavior of a modified predator–prey system with time delay is investigated. By regarding the time delay as the bifurcation parameter, we study the impact of the time delay on the dynamics of the system. The analysis shows that Hopf bifurcation can occur as the time delay passes some critical values. By employing the normal form theory and the center manifold reduction for functional differential equation, some sufficient conditions are derived for the direction and stability of the Hopf bifurcation. Finally, to verify our theoretical predictions, some numerical simulations are also included.

Dynamics in a diffusive predator–prey system with strong Allee effect and Ivlev-type functional response

The dynamics of a kind of reaction–diffusion predator–prey system with strong Allee effect in the prey population is considered. We prove the existence and uniqueness of the solution and give a priori bound. Hopf bifurcation and steady state bifurcation are studied. Results show that the Allee effect has significant impact on the dynamics.

Periodic traveling wave train and point-to-periodic traveling wave for a diffusive predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type scheme is investigated in this article. The existences of a small amplitude periodic traveling wave train Γp and the traveling wave solution connecting the boundary equilibrium Eu(1,0) to the periodic traveling wave Γp are obtained. The existence of this point-to-periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain, and thus plays a mild role in the evolution of predator–prey communities. The techniques used here are the Hopf bifurcation theorem, the improved shooting method combining with the geometric singular perturbation method.

Qualitative analysis of an Ivlev-type bio-economic system

The paper considers a bio-economic system with Ivlev-type functional response. Formally, our bio-economic system takes the form of differential-algebraic equations. We investigate Hopf bifurcation and center stability of the proposed bio-economic system. Some easily verifiable criteria are obtained on these issues. Finally, a numerical example is supplied to illustrate the effectiveness of our criteria.

Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.

The dynamical behaviors of a Ivlev-type two-prey two-predator system with impulsive effect

In this paper, considering the strategy of integrated Pest Management (IPM), a class of two-prey two-predator system with the Ivlev-type functional response and impulsive effect at different fixed time is established. By using impulsive comparison theorem, Floquent theory and small amplitude perturbation skill, the sufficient conditions for the system to be extinct of prey and permanence are proved. Moreover, we give two sufficient conditions for the extinction of one of two prey and remaining three species are permanent. Numerical simulation shows that there exist complex dynamics for system, such as symmetry-breaking pitchfork bifurcation, periodic doubling bifurcation, chaos, periodic halving cascade. Lastly, a brief discussion is given.

Dynamics of a marine plankton system: Diffusive instability and pattern formation

The paper deals with the study of the stability behavior of a marine prey–predator (phytoplankton–zooplankton) system. The interaction of the phytoplankton with zooplankton is taken as the Ivlev type functional response. The stability behaviors of the reaction–diffusion system arising from the zooplankton–phytoplankton model are carried out. We have studied the reaction–diffusion equation with both constant and variable (periodic) diffusion coefficients. For constant diffusion we have analyzed the effect of diffusion for diffusive instability of the system. The temporal variation of the system with small periodicity of diffusive coefficient have been studied by using Hill’s equation. We have carried out the numerical simulation to support and supplement the analytical results.

A sufficient and necessary condition for the existence of positive solutions for a prey–predator system with Ivlev-type functional response

We deal with a predator–prey interaction model with Ivlev-type functional response which is subject to the homogeneous Dirichlet boundary condition. On the basis of the fixed point index theory, we give a sufficient and necessary condition for the existence of positive solutions of the model.

Dispersal permanence of periodic predator–prey model with Ivlev-type functional response and impulsive effects

In this paper, we study the permanence of a periodic Ivlev-type predator–prey system where the prey disperses in patchy environment with two patches. We assume the Ivlev-type functional response within-patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent. Furthermore, we give numerical analysis to confirm our theoretical results. It will be useful to ecosystem control.