Toxin-determined Functional Response Research Articles

Spatiotemporal patterns of a diffusive plant–herbivore model with toxin-determined functional responses: Multiple bifurcations

In this paper, a homogeneous diffusive plant–herbivore model with toxin-determined functional response subject to the homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. By using Hopf bifurcation theorem and steady state bifurcation theorem due to Yi et al. (2009), we are able to show the existence of Hopf bifurcating periodic solutions (spatially homogeneous and non-homogeneous) and bifurcating non-constant steady state solutions. In particular, under certain conditions, the globally asymptotic stability of the positive constant steady state solutions and the non-existence of non-constant positive steady state solutions are investigated. These results allow the clear understanding of the mechanisms of the spatiotemporal pattern formations of this ecology model. In particular, our numerical results authenticate that toxicant parameter will play important roles in the stability and instability of the periodic solutions.

带有毒素效应功能反应的植物–食植动物模型的脉冲控制研究

带有毒素效应功能反应的植物–食植动物模型的脉冲控制研究

Persistence of pollination mutualisms under pesticides

A plant–pollinator system with pesticides is considered, in which the plant or plant’s seeds are treated by pesticides and the food (e.g., pollen and nectar) produced by plant is contaminated. The effect of pesticides is characterized by a toxin-determined functional response, which is a modification of the traditional Beddington–DeAngelis functional response through explicitly including a reduction in the consumption of food by the pollinator due to pesticides. Dynamics of the model demonstrate basic mechanisms by which the toxicity results in reduction of pollinators and leads to extinction of pollination mutualisms. It is also shown that varying the toxicity level could make the dynamics transition between uniform persistence, bi-stability, tri-stability and species’ extinction in a smooth fashion. Moreover, a detailed bifurcation analysis of the system reveals a rich array of novel behaviors including double saddle-node bifurcation and double transcritical bifurcation, which exhibits important biological implications in the complex dynamics of pollination mutualisms under pesticides.

Spatial distribution of microalgae in marine systems: A reaction–diffusion model

In this paper, we have proposed a reaction–diffusion system of partial differential equations which model the plankton-nutrient interaction mediated by a toxin-determined functional response. It has been established that microalgae, a clean and green source of energy, can be potentially used for carbon capture and sequestration. The common biofuels (bio-diesel and ethanol) are efficiently extracted from microalgae of different shapes and sizes. A spatio-temporal model has been presented to guide exploration and harvesting of microalgae (e.g., dinoflagellates, cilliates, chlorella, etc.). The spatial distribution of the phytoplankton (microalgae) is determined by growth pattern of the biotic subsystem (phytoplankton and zooplankton); e.g., whether it is oscillatory or aperiodic. The model incorporates a toxin-determined functional response of the zooplankton, which can be parametrized for specific phytoplankton–zooplankton combinations in different aquatic bodies such as ponds, seas, and oceans. The present model does not take into account higher zooplankton’s role in maintaining the core subsystem. The temporal model is analytically investigated in terms of the existence criteria and stability analysis (both linear and nonlinear) of the possible equilibria and the spatio-temporal model is studied in terms of global stability, Turing instability and existence of Hopf-bifurcation which help us to explore the dynamical behavior of the spatial model system. Numerical simulations are carried out to support the obtained theoretical results. Simulation experiments and computed densities thereof (equal densities are codes by same color) suggest that the spatial distribution of microalgae is complex; e.g., spatial density of microalgae varies chaotically for certain parameter sets. Harvesting schedule can be designed based on information thus derived. It should be implemented carefully in case the spatial density distribution is chaotic.The sustainability of the marine system for future use has been the prime concern. Parameters of harvesting strategy (time, intensity and technology) are determined in such a way that exploitation causes minimal damage to the environment and the yield of the harvest is maximal. Future studies would consider larger carnivorous fishes (e.g., Squids, Dolphins) on system’s dynamics. The effect of oceanic noise and colloidal swarming of zooplankton in the presence of bacteria will also be incorporated.

Plankton nutrient interaction model with effect of toxin in presence of modified traditional Holling Type II functional response

A three-dimensional plankton-nutrient interaction model is proposed and analysed which mediated by a toxin-determined functional response. The new functional response is a modification of the traditional Holling Type II functional response by explicitly including a reduction in the consumption of phytoplankton by the zooplankton due to chemical defenses. Our analysis leads to different thresholds in terms of model parameters to find out different steady-states behaviour. It is found that constant nutrient input and dilution rate of nutrient influence the plankton ecosystem model and maintain stability around the coexistent equilibrium. Our observations indicate that if the constant nutrient input crosses a certain critical value, the system enters into Hopf bifurcation. In addition, we have studied the direction of Hopf bifurcation by applying the normal form method. The maximal amount of toxin of phytoplankton species plays a crucial role to change the steady-state behaviour. Computer simulations illustrate the results.

Algae-herbivore interactions with Allee effect and chemical defense

Macroalgae exhibit a variety of characteristics that provide a degree of protection from herbivores. One characteristic is the production of chemicals that are toxic to herbivores. The toxic effect of macroalgae on herbivorous reef fish is studied by means of a spatiotemporal model of population dynamics with a nonmonotonic toxin-determined functional response of herbivores. It is assumed that the growth rate of macroalgae is mediated by Allee effect. We see that under certain conditions the system is uniformly persistent. Conditions for local stability of the system is obtained with weak and strong Allee effects. We observe that in presence of Allee effect on macroalgae, the system exhibits complex dynamics including Hopf bifurcation and saddle-node bifurcation. The obtained results show that the spatiotemporal system does not exhibit diffusion-driven instability. Computer simulations have been carried out to illustrate different analytical results.

Formation of time patterns in a diffusive plant–herbivore system with toxin-determined functional response

In this paper, we consider a diffusive plant–herbivore system with the toxin-determined functional response and subject to the homogeneous Neumann boundary conditions on the bounded one-dimensional spatial domain. The impacts of diffusion on the dynamical behaviors are investigated and it is found that although the appearance of diffusion does not affect the stability of constant steady states, it can lead to the occurrence of Hopf bifurcation of spatially inhomogeneous periodic solutions at the constant positive steady state. The conclusions show that the occurrence of spatial diffusion can induce more complex dynamical behaviors.

Existence of limit cycles and homoclinic bifurcation in a plant–herbivore model with toxin-determined functional response

In this paper we study a two-dimensional toxin-determined functional response model (TDFRM). The toxin-determined functional response explicitly takes into consideration the reduction in the consumption of plants by herbivore due to chemical defense, which generates more complex dynamics of the plant–herbivore interactions. The purpose of the present paper is to analyze the existence of limit cycles and bifurcations of the model. By applying the theories of rotated vector fields and the extended planar termination principle, we establish the conditions for the existence of limit cycles and homoclinic loop. It is shown that a limit cycle is generated in a supercritical Hopf bifurcation and terminated in a homoclinic bifurcation, as the parameters vary. Analytic proofs are provided for all results, which generalize the results presented in [11].

Dynamics in a diffusive plant–herbivore model with toxin-determined functional response

A diffusive plant–herbivore model with the toxin-determined functional response and homogeneous Neumann boundary condition is considered. The stability and Hopf bifurcation of the constant positive steady state are investigated by using the linearization method.

Periodic Solutions of a Stage-Structured Plant-Hare Model with Toxin-Determined Functional Responses

The purpose of this paper is to obtain some sufficient conditions for the global existence of multiple positive periodic solutions of a delayed stage-structured plant-hare model with a toxin-determined functional response. Some novel estimation techniques to construct two open subsets for a priori bounds are employed.

Multiple Periodic Solutions of a Nonautonomous Plant-Hare Model

Based on Mawhin's coincidence degree theory, sufficient conditions are obtained for the existence of at leasttwopositive periodic solutions for a plant-hare model with toxin-determined functional response (nonmonotone). Some new technique is used in this paper, because standard arguments in the literature are not applicable.

Spatially heterogeneous invasion of toxic plant mediated by herbivory

Spatially homogeneous (ODE) and reaction-diffusion models for plant-herbivore interactions with toxin-determined functional response are analyzed. The models include two plant species that have different levels of toxicity. The plant species with a higher level of toxicity is assumed to be less preferred by the herbivore and to have a relatively lower intrinsic growth rate than the less toxic plant species. Two of the equilibrium points of the system representing significant ecological interests are E1, in which only the less toxic plant is present, and E2, in which the more toxic plant and herbivore coexist while the less toxic plant has gone to extinction. Under certain conditions it is shown that, for the spatially homogeneous system all solutions will converge to the equilibrium E2, whereas for the reaction-diffusion model there exist traveling wave solutions connecting E1 and E2.

Global Dynamics of a Plant-Herbivore Model with Toxin-Determined Functional Response

Studies of the dynamics of plant-herbivore interactions that explicitly address model robustness are important in assessing uncertainty. Hence, identifying conditions that guarantee the global stab...

Dynamics of a plant–herbivore–predator system with plant-toxicity

A system of ordinary differential equations is considered that models the interactions of two plant species populations, an herbivore population, and a predator population. We use a toxin-determined functional response to describe the interactions between plant species and herbivores and use a Holling Type II functional response to model the interactions between herbivores and predators. In order to study how the predators impact the succession of vegetation, we derive invasion conditions under which a plant species can invade into an environment in which another plant species is co-existing with a herbivore population with or without a predator population. These conditions provide threshold quantities for several parameters that may play a key role in the dynamics of the system. Numerical simulations are conducted to reinforce the analytical results. This model can be applied to a boreal ecosystem trophic chain to examine the possible cascading effects of predator-control actions when plant species differ in their levels of toxic defense.

Dynamics of a plant-herbivore model with toxin-induced functional response

Traditional functional responses for plant-herbivore interactions do not take into account explicitly the effect of plant toxin. However, considerable evidence suggests that toxins set upper limits on food intake for many species of herbivorous vertebrates. In this paper, a mathematical model for plant-herbivore interactions mediated by toxin-determined functional response is studied. The model consists of three ordinary differential equations describing one herbivore population and two plant species with different toxicity levels. The effect of plant toxicity on herbivore's intake rate is incorporated explicitly in the model by assuming an increased handling time. The dynamical behaviors of the model are analyzed and the results are used to examine the influence of toxin-determined intake in the community composition of plant species. The bifurcation analysis presented in this paper suggests that the toxin-mediated functional response may have dramatic effects on plant-herbivore interactions.

Plant–herbivore interactions mediated by plant toxicity

We explore the impact of plant toxicity on the dynamics of a plant–herbivore interaction, such as that of a mammalian browser and its plant forage species, by studying a mathematical model that includes a toxin-determined functional response. In this functional response, the traditional Holling Type 2 response is modified to include the negative effect of toxin on herbivore growth, which can overwhelm the positive effect of biomass ingestion at sufficiently high plant toxicant concentrations. Two types of consumption decisions of the herbivore are considered. One of these (Case 1) incorporates the adaptation of the herbivore to control its rate of consumption of plant items when that is likely to lead to levels of toxicity that more than offset the marginal gain to the herbivore of consuming more plant biomass, while the other (Case 2) simply assumes that, although the herbivore’s rate of ingestion of plant biomass is negatively affected by increasing ingestion of toxicant relative to the load it can safely deal with, the herbivore is not able to prevent detrimental or even lethal levels of toxicant intake. A primary result of this work is that these differences in behavior lead to dramatically different outcomes, summarized in bifurcation diagrams. In Case 2, a wide variety of dynamics may occur due to the interplay of Holling Type 2 dynamics and the effect of the plant toxicant. These dynamics include the occurrence of bistability, in which both a periodic solution and the herbivore-extinction equilibrium are attractors, as well the possibility of a homoclinic bifurcation. Whether the herbivore goes to extinction in the bistable case depends on initial conditions of herbivore and plant biomasses. For relatively low herbivore resource acquisition rates, the toxicant effect increases the likelihood of ‘paradox of enrichment’ type limit cycle oscillations, but at higher resource acquisition rates, the toxicant may decrease the likelihood of these cycles.

Bifurcation analysis of a plant–herbivore model with toxin-determined functional response

A system of ordinary differential equations is considered which models the plant–herbivore interactions mediated by a toxin-determined functional response. The new functional response is a modification of the traditional Holling Type II functional response by explicitly including a reduction in the consumption of plants by the herbivore due to chemical defenses. A detailed bifurcation analysis of the system reveals a rich array of possible behaviors including cyclical dynamics through Hopf bifurcations and homoclinic bifurcation. The results are obtained not only analytically but also confirmed and extended numerically.