The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system

In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in a diffusive system with memory and general delays. The developed algorithm for calculating the normal form can be used to investigate the direction and stability of Hopf bifurcation. Then, we consider a diffusive predator-prey model with ratio-dependent Holling type-3 functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without considering gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. Furthermore, by using the developed algorithm for calculating the normal form, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found theoretically. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.

A Model for Pest Control using Integrated Approach: Impact of Latent and Gestation Delays

In this article, we have established a mathematical model using impulsive differential equations for the dynamics of crop pest management in the presence of a pest with its predator and bio-pesticides. The pest population is divided into two subpopulations, namely, the susceptible pests and the infected pests. In this control process, bio-pesticides (generally virus) infect the susceptible pest through viral infection within the pest and make it infected so that predators can consume it quickly. We assume that pest controlling, using this integrated approach, is a delayed process and thus incorporated latent time of susceptible pest and gestation delay of predator in the model as time delay parameters. The system dynamics have been analyzed using qualitative theory: the existence of the equilibrium points and their stability properties has been derived. Hopf bifurcation of the coexisting equilibrium point is presented for both the delayed and non-delayed system. Detail numerical simulations are performed in support of analytical results and illustrate the different dynamical regimes observed in the system. We have observed that the system becomes free of infection when the latent time of the pest is large. Coexisting equilibrium exists for the lower value of latent delay, and it can change the stability properties from stable to unstable when it crosses its critical value. In contrast, gestation delay affects the stability switches of coexisting equilibrium only. The combined effect of the two delays on the system is shown numerically. Also, viral replication rate, infection rate (from virus to pest) is also significant from the pest management perspective. In summary, both the delay is essential for crop pest management, and pest control will be successful with tolerable delays.

Spatial pattern formation and delay induced destabilization in predator-prey model with fear effect

Recent field experiments showed that predators influence the prey population not only by direct consumption but also by stimulating various defensive strategies. The cost of these defensive strategies can include energetic investment in defensive structures, reduced energy income, lower mating success, and emigration which ultimately reduces the reproduction of prey. To explore the effect of these defensive strategies (anti-predator behaviors), a modified Leslie-Gower predator-prey model with the cost of fear has been considered. Gestation delay is also incorporated in the system for a more realistic formulation. Boundedness, equilibria and stability analysis of the temporal model are studied. By considering gestation delay as a bifurcation parameter, the existence of Hopf-bifurcation around the interior equilibrium point is discussed together with the direction, stability and period of bifurcating solutions arising through Hopf-bifurcation. The spatial extension of the proposed model incorporating density-dependent cross-diffusion is also investigated and the conditions for diffusion-driven instability are obtained. To illustrate the analytical findings, detailed numerical simulations are performed. Biologically realistic Turing patterns as hexagonal spots, spots and stripes mixture, and labyrinthine type patterns are identified. It is found that the fear level has a stabilizing impact on delay induced destabilization and both stabilizing and destabilizing effects on Turing instability.

Dynamics of prey–predator model with strong and weak Allee effect in the prey with gestation delay

This study proposes two prey–predator models with strong and weak Allee effects in prey population with Crowley–Martin functional response. Further, gestation delay of the predator population is introduced in both the models. We discussed the boundedness, local stability and Hopf-bifurcation of both nondelayed and delayed systems. The stability and direction of Hopfbifurcation is also analyzed by using Normal form theory and Center manifold theory. It is shown that species in the model with strong Allee effect become extinct beyond a threshold value of Allee parameter at low density of prey population, whereas species never become extinct in weak Allee effect if they are initially present. It is also shown that gestation delay is unable to avoiding the status of extinction. Lastly, numerical simulation is conducted to verify the theoretical findings.

Modeling the Effect of Fear in a Prey–Predator System with Prey Refuge and Gestation Delay

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

Bifurcation in Plant-Pest-Natural Enemy Interaction Dynamics with Gestation Delay for Both Pest and Natural Enemy

During this analysis, as per natural control approach in pest management, a plant-pest dynamics with biological control is proposed, here assuming that the pest and natural enemy are having different levels of gestation delay and harvesting rate of pests by natural enemy follows Holling type-III response function. Boundedness and positivity of the system are studied. Equilibria and stability analysis is carried out for possible equilibrium points. The existence of Hopf bifurcation at interior equilibrium is presented. The sensitivity analysis of the system at interior equilibrium point for model parameters has been explored. Numerical simulations are performed to support our analytic findings.

Analysis and optimization based on a sex pheromone and pesticide pest model with gestation delay

Sex pheromone, aiming at mating disruption (MD), being species specific and leaving no toxic residues in the produce grown, offers an attractive alternative to conventional pesticides. In this paper, by incorporating the gestation delay and sex pheromone, we explore the impact of MD control on the dynamic behaviors of pest system. Firstly, the boundness, stability and bifurcation of system are deliberated. Secondly, an optimal control problem based on sex pheromone and pesticide is transformed into an equivalent optimal parameter selection problem by introducing the constrain violation function. Additionally, the gradients of the cost function with respect to the dose of sex pheromone and the killing rate are given. Furthermore, simulations are executed to validate the validity of our method. Meanwhile, our results indicate that gestation delay increases the extinction risk of the population and liberating sex pheromone destroys the stability of equilibrium states.