Dynamics of additional food provided predator–prey system with mutually interfering predators

Importance of pesticide and additional food in pest-predator system: a theoretical study

Integrated pest management (IPM) combines chemical and biological control to maintain pest populations below economic thresholds. The impact of providing additional food for predators on pest-predator dynamics, along- side pesticide use, in the IPM context remains unstudied. To address this issue, in this work a theoretical model was developed using differential equations, assuming Holling type II functional response for the predator, with additional food sources included. Strategies for controlling pest populations were derived by analyzing Hopf bifurcation occurring in the system using dynamical system theory. The study revealed that the quality and quantity of additional food supplied to predators play a crucial role in the system's dynamics. Pesticides, combined with the introduction of predators supported by high-quality supplementary food, enable a quick elimination of pests from the system more effectively. This observation highlights the role of IPM in optimizing pest management strategies with minimal pesticide application and supporting the environment.

Cannibalistic enemy-pest model: effect of additional food and harvesting

Biological control using natural enemies with additional food resources is one of the most adopted and ecofriendly pest control techniques. Moreover, additional food is also provided to natural enemies to divert them from cannibalism. In the present work, using the theory of dynamical system, we discuss the dynamics of a cannibalistic predator prey model in the presence of different harvesting schemes in prey (pest) population and provision of additional food to predators (natural enemies). A detailed mathematical analysis and numerical evaluations have been presented to discuss the pest free state, coexistence of species, stability, occurrence of different bifurcations (saddle-node, transcritical, Hopf, Bogdanov-Takens) and the impact of additional food and harvesting schemes on the dynamics of the system. It has been obtained that the multiple coexisting equilibria and their stability depend on the additional food (quality and quantity) and harvesting rates. Interestingly, we also observe that the pest population density decreases immediately even when small amount of harvesting is implemented. Also the eradication of pest population (stable pest free state) could be achieved via variation in the additional food and implemented harvesting schemes. The individual effects of harvesting parameters on the pest density suggest that the linear harvesting scheme is more effective to control the pest population rather than constant and nonlinear harvesting schemes. In the context of biological control programs, the present theoretical work suggests different threshold values of implemented harvesting and appropriate choices of additional food to be supplied for pest eradication.

A phytoplankton-zooplankton-fish model with chaos control: In the presence of fear effect and an additional food

The interplay of phytoplankton, zooplankton, and fish is one of the most important aspects of the aquatic environment. In this paper, we propose to explore the dynamics of a phytoplankton-zooplankton-fish system, with fear-induced birth rate reduction in the middle predator by the top predator and an additional food source for the top predator fish. Phytoplankton-zooplankton and zooplankton-fish interactions are handled using Holling type IV and II responses, respectively. First, we prove the well-posedness of the system, followed by results related to the existence of possible equilibrium points. Conditions under which a different number of interior equilibria exist are also derived here. We also show this existence numerically by varying the intrinsic growth rate of phytoplankton species, which demonstrates the model's vibrant nature from a mathematical point of view. Furthermore, we performed the local and global stability analysis around the above equilibrium points, and the transversality conditions for the occurrence of Hopf bifurcations and transcritical bifurcations are established. We observe numerically that for low levels of fear, the system behaves chaotically, and as we increase the fear parameter, the solution approaches a stable equilibrium by the route of period-halving. The chaotic behavior of the system at low levels of fear can also be controlled by increasing the quality of additional food. To corroborate our findings, we constructed several phase portraits, time-series graphs, and one- and two-parametric bifurcation diagrams. The computation of the largest Lyapunov exponent and a sketch of Poincaré maps verify the chaotic character of the proposed system. On varying the parametric values, the system exhibits phenomena like multistability and the enrichment paradox, which are the basic qualities of non-linear models. Thus, the current study can also help ecologists to estimate the parameters to study and obtain such important findings related to non-linear PZF systems. Therefore, from a biological and mathematical perspective, the analysis and the corresponding results of this article appear to be rich and interesting.

An Eco-Epidemiological Model Incorporating Harvesting Factors

The biological system relies heavily on the interaction between prey and predator. Infections may spread from prey to predators or vice versa. This study proposes a virus-controlled prey-predator system with a Crowley–Martin functional response in the prey and an SI-type in the prey. A prey-predator model in which the predator uses both susceptible and sick prey is used to investigate the influence of harvesting parameters on the formation of dynamical fluctuations and stability at the interior equilibrium point. In the analytical section, we outlined the current circumstances for all possible equilibria. The stability of the system has also been explored, and the required conditions for the model’s stability at the equilibrium point have been found. In addition, we give numerical verification for our analytical findings with the help of graphical illustrations.

An Optimal Control Study with Quantity of Additional food as Control in Prey-Predator Systems involving Inhibitory Effect

Additional food provided prey-predator systems have become a significant and important area of study for both theoretical and experimental ecologists. This is mainly because provision of additional food to the predator in the prey-predator systems has proven to facilitate wildlife conservation as well as reduction of pesticides in agriculture. Further, the mathematical modeling and analysis of these systems provide the eco-manager with various strategies that can be implemented on field to achieve the desired objectives. The outcomes of many theoretical and mathematical studies of such additional food systems have shown that the quality and quantity of additional food play a crucial role in driving the system to the desired state. However, one of the limitations of these studies is that they are asymptotic in nature, where the desired state is reached eventually with time. To overcome these limitations, we present a time optimal control study for an additional food provided prey-predator system involving inhibitory effect with quantity of additional food as the control parameter with the objective of reaching the desired state in finite (minimum) time. The results show that the optimal solution is a bang-bang control with a possibility of multiple switches. Numerical examples illustrate the theoretical findings. These results can be applied to both biological conservation and pest eradication.

Traveling wave phenomena in a nonlocal dispersal predator-prey system with the Beddington-DeAngelis functional response and harvesting

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $c^*$ such that the system possesses a traveling wave solution for any given $c> c^*$. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $c=c^*$ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $c<c^*$ is also discussed.

The impact of additional food on plankton dynamics in the absence and presence of toxicity

This article represents a three-dimensional dynamical system which studies the impact of additional food on the plankton system in the absence and presence of toxicity. It is determined that available additional food and toxin produced by phytoplankton play a crucial role in the termination of harmful algal blooms. The significance of the parameter, additional food, lies in the fact that its limited availability can terminate planktonic blooms when the concentration of toxicity is high. The positivity, boundedness, and permanence of the system are proved to make the model biologically valid. The existence of different biological feasible equilibria and their local stability criteria are determined analytically under certain conditions. The Hopf-bifurcation analysis is carried out by considering additional food as a bifurcation parameter. All the theoretical results are verified numerically. Our results indicate that the prey and predator population are never extinct and survive at a stable level when the additional food is available for the predator population.

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.