Holling Type Research Articles

A role of fear on diseased food web model with multiple functional response.

In this paper, we analyze the role of fear in a three-species non-delayed ecological model that examines the interactions among susceptible prey, infectious (diseased) prey, and predators within a food web. The prey population grows in a logistic manner until it achieves a carrying capacity, reflecting common population dynamics in the absence of predators. Diseased prey is assumed to transmit infection to healthful prey by the use of a Holling type II reaction. Predators, alternatively, are modeled to consume their prey using Beddington--DeAngelis and Crowley--Martin response features. This evaluation specializes in ensuring the non-negativity of solutions, practical constraints on population dynamics, and long-term stability of the system. Each biologically possible equilibrium point is tested to understand the environmental stable states. Local stability is assessed through eigenvalue analysis, while global stability of positive equilibria is evaluated by the use of Lyapunov features to determine the overall stability of the model. Furthermore, Hopf bifurcation is explored primarily based on infection rate $\varepsilon$. Numerical simulations are carried out to validate the theoretical effects and offer practical insights into the model behaviour under specific conditions.

Impact of Honest Signal, Cues, Prey’s Experience Rate and Environmental Toxicity in a Predator–Prey Interaction Model

Biological signaling plays a crucial role in predator–prey interaction systems, serving both intraspecific and interspecific communication purposes. Prey populations utilize signaling internally, employing honest signals among themselves to enhance their defense capabilities. Conversely, they also use signaling for interspecific communication, emitting cues to predators to increase their own chances of survival. Recently, Al-Salman et al. [Appl. Math. Modell. 89, 1405 (2021)] proposed a two-species interaction model incorporating Holling Type II predation along with honest signals and cues. In this paper, we extend and modify their study by incorporating factors such as prey’s experience rate and environmental toxicity in prey populations. We conduct a comprehensive mathematical and numerical analysis of the model. We investigate the positive invariance and dissipativeness of the model, discuss the existence of equilibrium points and analyze the behavior of the system around all equilibrium points. Particularly, we determine conditions for local stability and Hopf bifurcation of the system. Furthermore, we perform a detailed numerical analysis to study periodic solutions through Hopf bifurcations in the proposed system. We plot two-parameter bifurcation diagrams to examine changes in the system with combined effects, identifying critical values for stability switching. There is no chaotic dynamics seen in our investigation. Numerical results indicate that the system tends to stabilize when the experience rate is high, and the rate of error-prone honest signals regulates system stability. Additionally, we analyze how environmental toxicity affects the stability of the system.

Ecological‐economic modeling of nutrient enrichment in multispecies fisheries

AbstractIn fisheries economics, multispecies or ecosystem models typically focus on interactions among fish species, often overlooking the flow of nutrients. However, biologists widely acknowledge that nutrients–Âİsuch as nitrogen, phosphorus, potassium, and calcium–Âİare vital for organisms and play crucial roles in ecosystem dynamics. This paper delves into a bioeconomic predator‐prey model incorporating nutrient enrichment through Holling Type II and Michaelis–Menten functional forms. Initially, we examine the system's dynamics before formulating an optimal time‐dependent harvest policy using the Pontryagin maximum principle. Our findings reveal that heightened nutrient enrichment can lead to the breakdown of a positive stable equilibrium, resulting in the extinction of both prey and predator. The stability of the ecosystem may also be influenced by the interplay of nutrient enrichment and predator harvest.

Spatiotemporal dynamics of toxin producing phytoplankton–zooplankton interactions with Holling Type II functional responses

In this paper, we have studied the spatiotemporal dynamics of phytoplankton-zooplankton interactions using toxin-producing phytoplankton (TPP) with Holling type II functional responses. Toxin-producing phytoplankton (TPP) diffuses and reduces the grazing pressure on zooplankton. The grazing pressure of zooplankton is lessened by toxin-producing phytoplankton (TPP). The temporal system identifies all equilibrium points. Boundedness and local stability are established under specific parametric conditions. The conditions for the existence of a Hopf-bifurcation at the positive equilibrium by taking the half-saturation constant (b1), are also discussed. In a spatial system, we explored Turing instability conditions and patterns with an emphasis on the effect of diffusion variation. Furthermore, we obtained the time evaluation pattern formation of the spatial system. Moreover, the communities of toxin-producing phytoplankton (TPP) are essential to the marine ecosystem because they lessen the mortality of zooplankton caused by grazing pressure. Finally, the basic outcomes are mentioned along with numerical results to provide some support to the analytical findings.

Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin incidence and Holling type II treatment rates

This article presents a time-delayed SIR epidemiological model that has been quantitatively examined. The model incorporates a logistic growth function for the susceptible population, a Crowley-Martin type incidence, and Holling type II treatment rates. We investigated two separate time delays. The first delay refers to the rate at which new infections occur, allowing us to evaluate the impact of the latent period. The second delay relates to the rate of treatment for those who have contracted the infection, which allows us to examine the consequences of postponed access to therapy. The investigation of the steady-state behavior of the model emphasizes two equilibria, namely, the infection-free equilibrium and the endemic equilibrium. The determination of critical values involves the use of the fundamental reproduction number, denoted R0, which serves as a predictive measure to determine the potential elimination of a disease within a specific population. Using the fundamental reproduction number, it can be shown that infection-free equilibrium exhibits local asymptotic stability when the value of R0 is less than 1. In contrast, when R0 exceeds 1, the infection-free equilibrium becomes unstable in the context of the time-delayed system. Furthermore, an analysis of the steady-state dynamics of the endemic equilibrium indicates the appearance of oscillations and periodic solutions with the Hopf bifurcation for all feasible combinations of two-time delays as the bifurcated parameter. In sensitivity analysis, a sensitivity index is utilized to evaluate the relative modification in the fundamental reproduction number caused by each parameter. In summary, numerical simulations are employed to offer empirical evidence for the theoretical findings.

Plankton interaction model: Effect of prey refuge and harvesting

Abstract Harmful algal blooms are one of the major threats to aquatic ecosystem. Some phytoplankton species produce toxins during algal bloom and affect other aquatic species as well as human beings. Thus, for the conservation of aquatic habitat, it is much needed to control such phenomenon. In the present study, we propose a mathematical model of toxin-producing phytoplankton and zooplankton species, which follows the Holling Type III functional response. We consider the effect of prey refuge and harvesting on both the species. Boundedness of the proposed model, existence of equilibria, and their stability have been discussed analytically. We also discuss the optimal harvesting policy and existence of bionomic equilibrium. The numerical simulation has also been performed. We identify the control parameters that are responsible for the system dynamics of the model. The parameter prey refuge has a great impact on the dynamics of the model system. Higher value of prey refuge leads to the stable dynamics. Also, the growth rate of phytoplankton acts as a control parameter for the dynamics of the model. The higher value of growth rate of phytoplankton is responsible for oscillatory behavior.

Predation Efficiency and Biological Control Potential of Micromus angulatus Against Aphis craccivora

Micromus angulatus (Neuroptera: Hemerobiidae) is a widely distributed and highly effective predator that shows promise as a biological control agent against agricultural pests, particularly Aphis craccivora, the cowpea aphid, which threatens leguminous crops globally. This study aimed to evaluate the predation behaviour, search efficiency, and intraspecific interference of M. angulatus at different developmental stages, including first- to third-instar larvae and adults, in controlling adult A. craccivora populations. The results demonstrated that all developmental stages of M. angulatus exhibited predatory behaviour towards adult aphids, with the functional response fitting the Holling Type II model. The instantaneous attack rates for first-, second-, and third-instar larvae and adults were 1.0017, 1.0448, 0.9581, and 0.9508, respectively; the handling times were 0.0158, 0.0051, 0.0016, and 0.0011 days, respectively; and the theoretical maximum daily predation rates were 63.2911, 196.0784, 625, and 909.0909 aphids, respectively. The pest control efficacies were 63.3989, 204.8672, 598.8311, and 864.3192, respectively. The search efficiency at each developmental stage was negatively correlated with aphid density, which decreased as the prey density increased, with second-instar larvae showing the greatest decrease and adults the least. When the aphid density was fixed, the daily predation rate of individual M. angulatus decreased with increasing conspecific density, indicating that predation was affected by its own density, with the interference effect equation being E = 0.6194P−0.87. These findings indicate that M. angulatus, especially in the third-instar larval and adult stages, has considerable potential as a biological control agent for managing A. craccivora populations in agricultural settings. This study contributes valuable insights for developing sustainable agricultural practices by decreasing reliance on chemical pesticides.

Stability analysis of a delayed fractional-order $$\mathcal {SIR}$$ epidemic model with Crowley–Martin type incidence rate and Holling type II treatment rate

Stability analysis of a delayed fractional-order $$\mathcal {SIR}$$ epidemic model with Crowley–Martin type incidence rate and Holling type II treatment rate

Coexistence of populations in a Leslie-Gower tritrophic model with Holling-type functional responses

The dynamics of a Leslie-Gower type tritrophic model are analyzed. The model considers the interaction among three populations, a resource, a predator and a superpredator. It is assumed that the predator is generalist and its interaction with the resource is according to a general Holling-type functional response. Furthermore, it is assumed that the superpredator is specialist and its interaction with the predator follows a Holling type II functional response. The goal of this work is to show conditions that guarantee the coexistence of the three species. To do this, the existence of a stable equilibrium point or a stable limit cycle is demonstrated, which appears via a bifurcation. In addition, the analytical results are exemplified through numerical simulations.

Canard cycle, relaxation oscillation and cross-diffusion induced pattern formation in a slow–fast ecological system with weak Allee effect

In this paper, we consider a Holling type IV functional response that describes a situation in which the predator’s per capita rate of predation decreases at sufficiently high prey density. Meanwhile it can be transformed into a Holling type II functional response in the limiting sense where predator’s attack rate increases at a decreasing rate with prey density until it becomes satiated. To explore the different dynamics of these two functional responses, we consider the slow–fast dynamic behavior of predator–prey system with Holling type IV and II functional responses, respectively, and make some simple comparisons of the dynamics of the system with these two functional responses. Specifically speaking, in the nonspatial case, firstly, the system with Holling type II functional response does not undergo a higher-codimension Hopf bifurcation. Then, the system with Holling type IV is extensively proved the existence of canard cycles and relaxation oscillations by using a range of analytical methods such as geometric singular perturbation technique, normal form of the slow–fast system, and the way in–way out function. In the spatial case, the temporal systems are extended to reaction–diffusion predator–prey systems. For the reaction–diffusion system with Holling IV, the different types of traveling wave are observed. Moreover, for the reaction–diffusion predator–prey system with Holling II, it is demonstrated that Turing instability occurs, which induces spatial heterogeneity patterns. Finally, the comparisons of the above dynamics of Holling Type IV and II functional responses in the non-spatial and spatial cases, respectively, are presented. From these comparisons, different Holling functional responses may be adopted for species at different stages or states, which is more conducive to maintaining species diversity and coexistence.

Modeling the differential functional responses and selectivity of a marine copepod to nano/microplastics in mixture

Nano- and microplastics (NMPs) pollution is widespread in the oceans, posing potential risks to marine species. This study examined the accumulation capacity and selectivity potentials of NMPs by a marine copepod Parvocalanus crassirostris under different food mixtures by modeling the combined biokinetic and functional response. We investigated two sizes of NMPs (200 nm and 5 µm) across a concentration gradient (0 - 5000 µg/L) and varying diatom abundances (0, 104, 105 cells/mL). Fluorescence imaging and quantification revealed that P. crassirostris actively ingested NMPs at low concentration. Accumulation increased with NMPs concentration but eventually saturated due to gut capacity limits, following a Holling type II functional response (i.e., hyperbolic curve). Our novel functional response model estimated the key parameters and demonstrated that the maximum accumulation reached 5.3 % of dry weight with averaged half-saturation constants of 229 µg/L. The size of NMPs did not significantly affect the total accumulation or satiety levels. The presence of diatoms influenced the feeding selectivity and decreased the microplastic accumulation by 73 % at 105 cells/mL, while facilitating nanoplastic accumulation by 81 % at 104 cells/mL. This study enhanced our understanding of NMPs bioavailability and environmental fate in marine ecosystems.

Global bifurcation analysis and derivation of an optimal harvesting policy of a prey–predator model with Holling type-IV functional scheme

ABSTRACT This paper establishes a prey–predator model with a Holling type IV functional response and proportional harvesting in prey species and nonlinear harvesting in predator species. We provide a completely local and global bifurcation analysis of the model to characterize some meaningful results that may emerge from the interaction of biological resources. The proposed model exhibits various complex behaviors involving heteroclinic and homoclinic orbits, and we prove analytically that it also undergoes transcritical, Hopf, and Bogdanov–Takens bifurcations in the prey–predator plane. Moreover, the system is unfolded to compute the bifurcation curves, highlighting the critical dynamics by perturbing two bifurcation parameters near the cusp. Furthermore, from the economic point of view, the bionomic equilibrium of the system is studied, and optimal harvesting policy is derived using the Pontryagin’s maximum principle. Numerical simulations are presented to verify our analytical results.

Dynamical Behaviour of a Predator-Prey System with Holling Type III Functional Response under Harvesting and Self-crowding

Dynamical Behaviour of a Predator-Prey System with Holling Type III Functional Response under Harvesting and Self-crowding

Nonlinear dynamics in a fear‐driven predator–prey system: Bistability, bifurcations, hydra effect, and optimal harvesting

The impact of predator‐driven fear on ecosystems is significant and can encompass both trophic (direct) and nontrophic (indirect) effects. Previous studies have shown that nontrophic fear effects have an important role in predator–prey dynamics. This study investigates the nontrophic fear effect on prey caused by generalist predators and explores optimal harvesting. We assume that the reproduction rate of prey is reduced due to the fear effect, and generalist predator follows Holling type II foraging strategy for predation. Additionally, we assume that predators are commercially valuable and harvested proportionately to their density. We demonstrate the existence of equilibrium points, their local and global stability, and bifurcation analysis. We observe that the model system undergoes a sequence of codimension one and codimension two bifurcations. Our results show that in the absence of predator harvesting, increasing levels of fear destabilize the predator–prey system, and controlled harvesting is beneficial for the coexistence of both populations. Also, the harvesting of predators may produce hydra and multiple hydra effects. We identify different types of bistability phenomena, which emphasize the importance of initial population size. Further, an optimal harvesting policy is also explored by formulating an optimal control problem (OCP). The harvesting cost‐functional is designed by incorporating the bionomic equilibrium state. We use Pontryagin's maximum principle and solve the OCP numerically. It is observed that implementing optimal harvesting not only contributes to the ecological benefits by maintaining a sustainable balance of predator–prey evolution but also plays a significant role in maximizing the economic benefits.

The dynamics in an intraguild prey-predator model with Holling type III functional response

The dynamics in an intraguild prey-predator model with Holling type III functional response

Identifying numerical bifurcation structures of codimensions 1 and 2 in interacting species system

The present study has focused on the examination of a Holling type III nonlinear mathematical model that incorporates the influence of fear and Michaelis–Menten‐type predator harvesting. The incorporation of fear with respect to the prey species has been observed to result in a reduction in the survival probability of the prey population and a concurrent reduction in the reproduction rate of the prey species. The existence and stability of ecologically significant equilibria have been ascertained through mathematical analysis. Emphasis within the proposed model primarily centers on numerical bifurcations of codimensions 1 and 2. Numerical validation has been performed on all simulated outcomes within the feasible range of parametric values. Dynamical characteristics of the model have subsequently undergone investigation through a series of numerical simulations, successfully revealing various forms of local and global bifurcations. In addition to the identification of saddle‐node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic, and limit point cycle (LPC) bifurcations, the model has also demonstrated bistability and global asymptotic stability. These bifurcation phenomena serve as illustrative examples of the intricate dynamical behavior inherent to the model. Numerical validation through graphical representations has been utilized to elucidate the effects of factors such as fear, nonlinear predator harvesting, and predation rate on the dynamics of the interacting species under different parametric conditions.

A dynamic analysis of capital, corruption, and labor market interactions

The complex relationship between corruption, economic growth, and labor dynamics has long been a challenge for policymakers. This study sheds new light on these dynamic interactions by constructing a novel economic model with saturation effects that integrate employment considerations. Unlike static or purely empirical models, our approach captures the system’s evolution over time, employing a logistic growth function to realistically reflect economic limitations and distinguishing between employed and unemployed labor segments. By incorporating a Holling type II function, the model captures the nonlinear influence of corruption on economic growth, acknowledging diminishing returns at higher corruption levels. Through advanced theoretical analysis, we establish the existence and stability conditions for three distinct long-term equilibria: stagnation, corruption-free growth with full employment, and a coexistent state where corruption persists alongside some economic activity. Our analysis goes beyond merely identifying these equilibria by revealing the specific conditions under which each state prevails, and we solidify these theoretical predictions with comprehensive numerical simulations depicting the system’s behavior under varying conditions. The findings underscore the importance of robust anti-corruption measures and a labor-intensive economic approach, highlighting that fostering growth outpacing resource depletion is crucial for escaping stagnation and achieving sustainable development. Overall, this research advances the field by integrating employment dynamics, a Holling type II function for corruption, and a focus on stability analysis, providing invaluable insights for policymakers aiming to promote sustainable economic development, combat corruption, and ensure healthy employment conditions.

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

Lack of impact of Wolbachia on foraging behavior and morphological characteristics of the parasitoid wasp, Habrobracon hebetor (Braconidae)

Wolbachia are the most widespread intracellular alphaproteobacteria in insects with a variety of phenoptypic effects on the fitness and reproduction of their host, but much less is known about how these bacteria affect host behavior. In this study, we asked whether Wolbachia affects the foraging behavior of the parasitoid wasp Habrobracon hebetor (Hym.: Braconidae), an important biological control agent of many lepidopteran larvae. To test this, we analyzed the functional and numerical responses of Wolbachia-infected and uninfected (tetracycline cured) wasps, as well as morphological parameters. Functional response analysis showed Holling type II responses in both the Wolbachia-infected and uninfected females. The handling time and searching efficiency of Wolbachia-infected and uninfected females were similar, although the estimated maximum parasitism rate was shorter in infected females. Regardless of Wolbachia infection status, there was a negative non-linear relationship between the number of larvae parasitized by a female and an increase in the host density, reflecting a decrease in the total number of eggs laid. Our results also showed that Wolbachia had no effect on morphological traits of the parasitoid wasp. Together, these results suggest that Wolbachia which are prevalent in H. hebetor have limited impact on foraging behavior and morphology.

On the stability of a Caputo fractional order predator-prey framework including Holling type-II functional response along with nonlinear harvesting in predator

In this study, we look at a fractional-order two-species predator-prey framework that uses Caputo Sense's fractional derivative to try to understand its dynamics, which include a simplified Holling type II functional response and nonlinear harvesting in the predator. Initially, we establish certain mathematical facts, such as the solutions of fractional order dynamical systems being unique, non-negative, bounded, and existing. The dissipativeness of the FDE system solution is addressed. Our investigation also includes a look at the local stability requirements for every possible equilibrium point. As a bifurcation constraint, the order of derivatives has been considered in our discussion of the existence condition of Hopf bifurcation. Our analysis has been strengthened by the incorporation of fractional Hopf bifurcation, and the dynamics of the proposed model are influenced by selective non-linear harvesting. Furthermore, the investigation focuses on the impact of the predator's pace of harvesting about the complexities of the prey-predator relationship. Through the use of certain setting values, empirical evidence indicates as the non-integer order framework displays stable to unstable, when the non-integer order is increased. The analytical findings are shown by various instances presented in the numerical part.