Tritrophic Food Chain Model Research Articles

Persistence and zero-Hopf equilibrium in the tritrophic food chain model with Holling functional response

In this paper, we analyze the persistence of three species in a three-level food chain model. We characterize when such a model exhibits a zero-Hopf equilibrium point and show that it is possible only if the functional responses in the model are of type Holling III or IV.

Zero‐Hopf bifurcation in a family of tritrophic food chain model with Holling III–III functional response

Zero‐Hopf bifurcation in a family of tritrophic food chain model with Holling III–III functional response

Chaos and extinction risks of sexually reproductive generalist top predator in a seasonally forced food chain system with Allee effect.

This paper investigates the dynamics of a tritrophic food chain model incorporating an Allee effect, sexually reproductive generalist top predators, and Holling type IV and Beddington-DeAngelis functional responses for interactions across different trophic levels. Analytically, we explore the feasible equilibria, their local stability, and various bifurcations, including Hopf, saddle-node, transcritical, and Bogdanov-Takens bifurcations. Numerical findings suggest that higher Allee intensity in prey growth leads to the inability of species coexistence, resulting in a decline in species density. Likewise, a lower reproduction rate and a higher strength of intraspecific competition among top predators also prevent the coexistence of species. Conversely, a rapid increase in the reproduction rate and a decrease in the strength of intraspecific competition among top predators enhance the densities of prey and top predators while decreasing intermediate predator density. We also reveal the presence of bistability and tristability phenomena within the system. Furthermore, we extend our autonomous model to its nonautonomous counterpart by introducing seasonally perturbed parameters. Numerical analysis of the nonautonomous model reveals that higher seasonal strength in the reproduction rate and intraspecific competition of top predators induce chaotic behavior, which is also confirmed by the maximum Lyapunov exponent. Additionally, we observe that seasonality may lead to the extinction of species from the ecosystem. Factors such as the Allee effect and growth rate of prey can cause periodicity in population densities. Understanding these trends is critical for controlling changes in population density within the ecosystem. Ecologists, environmentalists, and policymakers stand to benefit significantly from the invaluable insights garnered from this study. Specifically, our findings offer pivotal guidance for shaping future strategies aimed at safeguarding biodiversity and maintaining ecological stability amidst changing environmental conditions. By contributing to the existing body of knowledge, our study advances the field of ecological science, enhancing the comprehension of predator-prey dynamics across diverse ecological conditions.

Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory

The spatiotemporal dynamics of a three-component model of a food web are considered. The model describes the interactions between populations of resources, prey, and predators that consume both species. It assumes that the predator responds to the spatial change in the resource and prey densities by occupying areas where species density is higher (prey-taxis) and that the prey population avoids areas with a high predator density (predator-taxis). This work studies the conditions for the taxis-driven instability leading to the emergence of stationary patterns resulting from Turing instability and autowaves caused by wave instability. The existence of nonconstant positive steady states for the system is assessed through a rigorous bifurcation analysis. Meanwhile, the conditions for the existence of both types of instabilities are obtained by linear stability analysis. It is shown that the presence of cross-diffusion in the system supports the formation of spatially heterogeneous patterns. For low values of the resource-tactic and predator-tactic coefficients, Turing and wave instabilities coexist. The system undergoes only Turing instability for high levels of these parameters.

Bacteria as ecosystem engineers: Unraveling clues through a novel functional response and tritrophic model

Bacteria play a crucial role in regulating the nutrient cycle of ecosystems, making their abundance essential for the sustainability of these environments. Maintaining a thriving bacterial population is achieved through complex nutrient–bacteria–protozoa interactive dynamics, which can be likened to a tritrophic food chain model. The interaction between bacteria and protozoa is governed by a critical factor known as the bacteria–protozoa functional response, which serves as the foundation for developing this tritrophic model. However, existing functional responses have shown limitations in accurately describing the intricacies of the bacteria–protozoa interaction. One significant drawback is the neglect of bacterial behavioral traits. To address this issue, we consider the concept of cooperation as a group defense mechanism employed by bacteria facilitated through a quorum-sensing communication process. By incorporating the cooperation trait into the functional response, our model offers a more comprehensive understanding of the complex tritrophic food chain dynamics. We evaluate the stability of different equilibrium points, along with Hopf-bifurcation around the coexistence equilibrium point. We find that a balance between strong group defense and moderate cooperation is essential for bacteria sustainability and overall system stability. Our results also elaborately address the effects of the increasing group defense through the bistable equilibrium followed by a branch point and saddle–node bifurcation. Through comprehensive analyses and simulations, we examine the paradox of enrichment in nutrition flow at the community level and explore how nutrient washout controls system stability. This innovation not only enhances our comprehension of ecosystem sustainability but also opens up new avenues for studying the intricate relationships that govern the overall balance of nature.

An investigation of the parameter space in a tri-trophic food chain model with refuge

Aside from humans, the instinct to run and hide from dangerous situations is ingrained in individuals of many other species. Taking refuge is a very common precautionary strategy among prey species to avoid direct contact with their predators. In this article, we investigate a three-species food chain model where the logistically growing basal prey seeks refuge to avoid the middle predator. To assess the impact of varying levels of refuge, we explore the system dynamics in the biparametric space of refuge and another important system parameter related to the half-saturation constant of the basal prey. The isospike diagrams along with the largest Lyapunov exponent diagrams show a stabilizing role of refuge. Apart from this, we observe an abundance of organized periodic structures embedded inside the chaotic zone of the biparametric space. The most notable ones include the shrimp-shaped structures and an “elliptic disc” shaped structure. The article also explores the coexistence between multiple pairs of coexisting attractors and one instance of spike-bubbling phenomena. The basins of attraction for the coexisting attractors also have a self-similar pattern. We also modify the present model by replacing the density-dependent prey refuge with constant proportion of prey refuge, and then explore the dynamics of the modified model within the same biparametric space and find similar complexity in its dynamics. We compare the effects of both types of refuge on the density variations of all three species when they show stable coexistence and observe that density-dependent refuge is more beneficial to the system from the standpoint of species conservation.

Ecological Effects of Predator Harvesting and Environmental Noises on Oceanic Coral Reefs.

Coral reefs provide refuge for prey and are important for the preservation of an oceanic ecosystem. However, they have been experiencing severe destruction by environmental changes and human activities. In this paper, we propose and analyze a tri-trophic food chain model consisting of coral, Crown-of-thorns starfish (CoTS), and triton in deterministic and stochastic environments. We investigate the effects of harvesting in the deterministic system and environmental noises in the stochastic system, respectively. The existence of possible steady states along with their stability is rigorously discussed. From the economic perspective, we examine the existence of the bionomic equilibrium and establish the optimal harvesting policy. Subsequently, the deterministic system is extended to a stochastic system through nonlinear perturbation. The stochastic system admits a unique positive global solution initiating from the interior of the positive quadrant. The long-time behaviors of the stochastic system are explored. Numerical simulations are provided to validate and complement our theoretical results. We show that over-harvesting of triton is not beneficial to coral reefs and modest harvesting of CoTS may promote sustainable growth in coral reefs. In addition, the presence of strong noises can lead to population extinction.

Chaos Control of a Delayed Tri-Trophic Food Chain Model with Fear and Its Carry Over Effects

One of the main objectives of theoretical ecologists involves finding mechanisms to control the chaos in ecological models to maintain positive densities of the species. Numerous researchers have suggested that, apart from the direct killing in the prey–predator relationship, there are some indirect effects, such as fear of predation. Induced fear can lead to slowing down the growth rate of the prey species, and this non-chemical strategy can be carried over to successive seasons or upcoming generations. In this work, we explore the impact of fear due to predation and its carry-over effect (COE) in a delayed tri-trophic food chain model, whereas the Holling type-II functional response is used to determine the interference among the species. The proposed model is an asymmetric interaction food chain model since the species in this model only kills other species. The growth rate of prey and middle predators is affected due to the respective fear of predation by middle and special predators. The non-delayed model considered in this paper generalizes the models developed by Hastings–Powell and Panday et al. The gestation delay in the special predator’s growth term is incorporated into the proposed model. We determined the essential conditions for the existence of ecologically feasible equilibrium points and their local and global stability. Furthermore, we developed the conditions for the occurrence of the Hopf bifurcation around an interior equilibrium to seek periodic behaviors of delayed and non-delayed models. Numerical examples were performed to justify the proposed theoretical findings and to show the impacts of fear and its COE parameters on the system dynamics through phase portraits, the time series of solutions, and bifurcation diagrams. We discovered that the chaotic behavior of the food chain model can be controlled by using the fear effect and its COE parameters. The dynamics of the delayed food chain model with the fear effect and its COEs are further explored in our findings. Our theoretical findings clearly provide a mechanism to protect and control species populations in ecological systems. It is also essential for developing optimized harvesting strategies in fisheries and pest management in agriculture.

Impact of fear on a tri-trophic food chain model with supplementary food source

Impact of fear on a tri-trophic food chain model with supplementary food source

On the Dynamics of a Harvested Tri-trophic Food Chain Model with Alternative Food Source

On the Dynamics of a Harvested Tri-trophic Food Chain Model with Alternative Food Source

A chaotic tri-trophic food chain model supplemented by Allee effect

A chaotic tri-trophic food chain model supplemented by Allee effect

Stability and bifurcation analysis of a tri-trophic food chain model with intraguild predation

The dynamics of a three-component model for food web with intraguild predation is considered. The model is based on the collection of ordinary differential equations that describe the interactions among prey, intermediate predator and top predator. First, the model without self-limitation of the predators is studied. Boundedness of the system and existence of non-negative solutions are established. The local stability analysis of the equilibria is carried out to examine the behavior of the system. The possibility of Hopf bifurcation around non-negative equilibria with consumption rates as bifurcation parameters is studied. Center manifold theorem and the normal form theory are applied to obtain the formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations support the analytical findings, which show that the extinction of one of the predators can occur under certain restrictions on the predation rate of the top predator. Subsequently, numerical analysis of the model with self-limitation of the predators is carried out. Simulations reveal that the system with intraspecific competition in the predator populations can reproduce coexistence between the three species in resource-rich environment.

Untangling the role of tri-trophic food chain model in sustaining quokka population

The introduction of predators like red fox and dingoes in the 1930s took a big toll on the quokka population. This paper is mainly concerned with designing conservation policies for quokka population (an endangered species). For this purpose, we designed a reaction–diffusion tri-tropic food chain model consisting quokka and its two predators, red fox and dingoes. We have shown the global existence, non-negativity and uniform boundedness for the designed spatiotemporal model. Existence and stability analysis of the equilibrium points for the model is done. We applied the prevalent idea of basic reproduction number with its origin from epidemiology to the food chain model, to deduce a condition for extinction and persistence of predator population. Natural systems exhibit an amazing diversity of structures in both living and non-living systems. We found that in the presence of diffusion, the model has the potential of exhibiting Turing instability generating beautiful patterns. Numerical results reveal that quokka tends to avoid places resided by dingoes. Our research aims at finding a solution for the current quokka extinction problem by showing the effect of presence of alternative food for dingoes and prohibiting the external factor causing death of quokka population.

Controlling chaos and pattern formation study in a tritrophic food chain model with cannibalistic intermediate predator

Controlling chaos and pattern formation study in a tritrophic food chain model with cannibalistic intermediate predator

Deterministic and Stochastic Fractional-Order Hastings-Powell Food Chain Model

In this paper, a deterministic and stochastic fractional-order model of the tri-trophic food chain model incorporating harvesting is proposed and analysed. The interaction between prey, middle predator and top predator population is investigated. In order to clarify the characteristics of the proposed model, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions of the proposed model are examined. Some sufficient conditions that ensure the local and global stability of equilibrium points are obtained. By using stability analysis of the fractional-order system, it is proved that if the basic reproduction number , the predator free equilibrium point is globally asymptotically stable. The occurrence of local bifurcation near the equilibrium points is investigated with the help of Sotomayor’s theorem. Some numerical examples are given to illustrate the theoretical findings. The impact of harvesting on prey and the middle predator is studied. We conclude that harvesting parameters can control the dynamics of the middle predator. A numerical approximation method is developed for the proposed stochastic fractional-order model.

Explosive tritrophic food chain model with herd behaviour of prey and finite time blow-up of the top predator

Explosive tritrophic food chain model with herd behaviour of prey and finite time blow-up of the top predator

Explosive tritrophic food chain model with herd behaviour of prey and finite time blow-up of the top predator

Explosive tritrophic food chain model with herd behaviour of prey and finite time blow-up of the top predator

Dynamical complexity in a tritrophic food chain model with prey harvesting

The present investigation deals with a tritrophic food web model with Holling-Tanner type II functional response to clarify the dynamical complexity of the eco-systems in the natural environment. The objective of this study is to explore the harvesting mechanism scenario in a three-dimensional interacting species system such as one prey and two specialist predators. Attention has been given to demonstrate the system characteristics near the biologically feasible equilibria. Specifically, stability, Hopf-Andronov bifurcation for the respective system parameters and dissipativeness has been performed in order to scrutinize the system behaviour. Lyapunov exponents are worked out numerically and an unstable scenario for significant parameters of the model system has been executed to characterize the complex dynamics. In addition to, we put forward a detailed numerical simulation to justify the chaotic dynamics of the present system. We conclude that chaotic dynamics can be executed by the prey harvesting parameters.

Fear Effect in a Tri-Trophic Food Chain Model with Holling Type IV Functional Response

Of concern the present study deals with an updated food chain model comprising one prey and two predator species in a natural environment with the inclusion of fear effect in both the prey and the median predator population through Holling type IV functional response. The present model is affluent with intra-specific competition among the hunter species having specific mortality. The model system emphasizes its characteristics in the proximity of the probable equilibrium position in the realm of biological dynamics. The response of the system is explored further for its stability analysis based on prerequisites and Hopf-bifurcation phenomena as well with respect to some significant model parameters. A quantitative analysis based on numerical simulation is also carried out in order to validate the analytical results and thereby the applicability of the model is established.

A “Double” fear effect in a tri-trophic food chain model

A “Double” fear effect in a tri-trophic food chain model