Influence of consumer mutual interference on the stabilization of a three-level trophic system

In this paper, we present a three-level (food–prey–predator) trophic food chain which includes consumer mutual interference (MIF). In contrast with other analyses, we consider the effect of both prey and predator MIF on the dynamics of a three-level trophic system. MIF is generally considered to exert a stabilizing effect on population dynamics based on the predator–prey model. However, results from analytical and numerical simulations utilizing a simple three-species food chain model suggest that while the addition of prey MIF to the model provides a stabilizing influence, as the chaotic dynamics collapse to a stable steady state, adding only predator MIF to the model can only stabilize the system at intermediate MIF values. The three-species trophic food chain is also stabilized when combination of both prey and predator MIF is added to the model. Our work serves to provide insight into the effects of MIF in the real world.     

Chaos in three species food chains

We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model. This work extends earlier numerical studies of a particular system by Hastings and Powell (1991) by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by Guckenheimer and Holmes (1983) on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.     

Chaos in a periodically forced chemostat with algal mortality

We study the possibility of chaotic dynamics in the externally driven Droop model. This model describes a phytoplankton population in a chemostat under periodic nutrient supply. Previously, it has been proven under very general assumptions, that such systems are not able to exhibit chaotic dynamics. We show that the simple introduction of algal mortality may lead to chaotic oscillations of algal density in the forced chemostat. Our numerical simulations show that the existence of chaos is intimately related to plankton overshooting in the unforced model. We provide a simple measure, based on stability analysis, for estimating the amount of overshooting. These findings are not restricted to the Droop model but also hold for other chemostat models with mortality. Our results suggest periodically driven chemostats as a simple model system for the experimental verification of chaos in ecology.     

A detailed study of the Beddington-DeAngelis predator-prey model

The present investigation accounts for the influence of intra-specific competition among predators in the original Beddington-DeAngelis predator-prey model. We offer a detailed mathematical analysis of the model to describe some of the significant results that may be expected to arise from the interplay of deterministic and stochastic biological phenomena and processes. In particular, stability (local and global) and bifurcation (Saddle-node, Transcritical, Hopf-Andronov, Bogdanov-Takens) analysis of this model are conducted. Corresponding results from previous well known predator-prey models are compared with the current findings. Nevertheless, we also allow this model in stochastic environment with the influences of both, uncorrelated “white” noise and correlated “coloured” noise. This showing that competition among the predator population is beneficial for a number of predator-prey models by keeping them stable around its positive interior equilibrium (i.e. when both populations co-exist), under environmental stochasticity. Comparisons of these findings with the results of some earlier related investigations allow the general conclusion that predator intra-species competition benefits the predator-prey system under both deterministic and stochastic environments. Finally, an extended discussion of the ecological implications of the analytical and numerical results concludes the paper.     

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.     

Disease in prey population and body size of intermediate predator reduce the prevalence of chaos-conclusion drawn from Hastings–Powell model

In ecology the disease in the prey population plays an important role in controlling the dynamical behaviour of the system. We modify Hastings and Powell’s (HP) [Hastings, A., Powell, T., 1991. Chaos in three-species food chain. Ecology 72 (3), 896–903] model by introducing disease in the prey population. The conditions for which the modified HP model system represents extinction, permanence or impermanence of population are worked out. The modified model is analyzed to obtain different conditions for which the system exhibits stability around the biologically feasible equilibria. Through numerical simulations we display that the modified system enters into stable solutions depending upon the force of infection in prey population as well as body size of intermediate predator. Our results demonstrate that disease in prey population and body size of intermediate predator are the key parameters for controlling the chaotic dynamics observed in original HP model.     

高阶相互作用对宿主-寄生群落动态的影响

物种间相互作用是影响生物群落稳定性和多样性的重要因素。基于Lotka-Volterra竞争模型,通过构建多宿主种群的种内和种间高阶相互作用模型,研究宿主种群的间接竞争效应对寄生群落动态的影响机制。为有效地揭示高阶作用对种群动态的影响,通过对比宿主-寄生群落的现象模型以及机制模型,利用机制模型产生的合理数据集对现象模型中高阶项的参数进行拟合,进而探讨了高阶相互作用在群落动态中的作用。结果显示,完整的高阶相互作用模型在描述多宿主-寄生系统的群落动态中表现最优,而直接相互作用模型对群落动态的描述相对较差,即同时考虑种间和种内的高阶相互作用模型更加符合机制模型所描述的群落动态。此外,种内高阶作用和种间高阶作用产生不对称效应,宿主间的种间高阶作用对群落产生的影响较种内高阶作用更为显著。该研究结果在一定意义上丰富了宿主-寄生生物群落的稳定性研究,为理解物种间相互作用的多样性研究提供了依据。     

Food chains in the chemostat: Relationships between mean yield and complex dynamics

Atritrophic food-chain chemostat model composed of a prey with Monod-type nutrient uptake, a Holling Type II predator and a Holling Type II exploited superpredator is considered in this paper. The bifurcations of the model show that dynamic complexity first increases and then decreases with the nutrient supplied to the bottom of the food chain. Extensive simulations prove that the same holds for food yield, i.e., there exists an optimum nutrient supply which maximizes mean food yield. Finally, a comparative analysis of the results points out that the optimum nutrient supply practically coincides with the nutrient supply separating chaotic dynamics from high-frequency cyclic dynamics. This reinforces the idea, already known for simpler models, that food yield maximization requires that the system behaves on the edge of chaos.     

Modelling of phytoplankton allelopathy with Monod–Haldane-type functional response—A mathematical study

In this paper, a three-tier model of phytoplankton, zooplankton and nutrient is considered and stability of different equilibrium points is analyzed along with Hopf-bifurcation around coexisting equilibrium point. Here, we have assumed toxication process as the guiding factor for bloom formation as well as its termination and this process is incorporated into our model by choosing the zooplankton grazing function as a Monod–Haldane function due to the phytoplankton toxicity. Extensive numerical simulations have been performed to validate the analytical findings and these simulation work reveal the chaotic oscillation exhibited by the model system for certain choice of the parameter values.     

Deterministic limits to stochastic spatial models of natural enemies

Stochastic spatial models are becoming an increasingly popular tool for understanding ecological and epidemiological problems. However, due to the complexities inherent in such models, it has been difficult to obtain any analytical insights. Here, we consider individual-based, stochastic models of both the continuous-time Lotka-Volterra system and the discrete-time Nicholson-Bailey model. The stability of these two stochastic models of natural enemies is assessed by constructing moment equations. The inclusion of these moments, which mimic the effects of spatial aggregation, can produce either stabilizing or destabilizing influences on the population dynamics. Throughout, the theoretical results are compared to numerical models for the full distribution of populations, as well as stochastic simulations.     

Rapid prey evolution and the dynamics of two-predator food webs

Traits affecting ecological interactions can evolve on the same time scale as population and community dynamics, creating the potential for feedbacks between evolutionary and ecological dynamics. Theory and experiments have shown in particular that rapid evolution of traits conferring defense against predation can radically change the qualitative dynamics of a predator–prey food chain. Here, we ask whether such dramatic effects are likely to be seen in more complex food webs having two predators rather than one, or whether the greater complexity of the ecological interactions will mask any potential impacts of rapid evolution. If one prey genotype can be well-defended against both predators, the dynamics are like those of a predator–prey food chain. But if defense traits are predator-specific and incompatible, so that each genotype is vulnerable to attack by at least one predator, then rapid evolution produces distinctive behaviors at the population level: population typically oscillate in ways very different from either the food chain or a two-predator food web without rapid prey evolution. When many prey genotypes coexist, chaotic dynamics become likely. The effects of rapid evolution can still be detected by analyzing relationships between prey abundance and predator population growth rates using methods from functional data analysis.     

Time delay can enhance spatio-temporal chaos in a prey–predator model

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.     

Cancer self remission and tumor stability-- a stochastic approach

The paper aims to express the spontaneous regression and progression of a malignant tumor system as a prey--predator like system. The model is a three dimensional deterministic system, consisting of tumor cells, hunting predator cells and resting predator cells. Local stability analysis is performed along with numerical simulations to support the analytical findings. Moreover, the deterministic model is extended to a stochastic one allowing random fluctuations around the positive interior equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically. The thresholds obtained from our study may be helpful to control the malignant tumor growth.     

Species coexistence and chaotic behavior induced by multiple delays in a food chain system

In this paper, we present a food chain system composed of three species (resource, consumer and predator). Food digested periods corresponding to consumer-eat-resource and predator-eat-consumer are introduced for more realistic consideration, which are called resource digested delay (RDD) and consumer digested delay (CDD), respectively. In order to explore the combined influence of multiple delays on population dynamics, two different scenarios were explored, i.e. the intrinsic growth rate of resource is less/more than consumer and predator. In case 1, some types of species coexistence characterized by RDD and CDD independent are illustrated by the corresponding characteristic equation. Multiple delays can promote and suppress the recurrent bloom of species population, which is called the stability switching of species coexistence. In case 2, using RDD and CDD as the varying parameters, complex dynamical behaviors including multiple periodic motion and chaotic behavior are exhibited in detail by employing some numerical simulations, such as phase trajectory, power spectra, and bifurcation diagram. The population dynamics exhibit chaos behavior and then evolve into system collapse for species outbreak. Further, greater RDD and CDD make system population enter into system collapse easier.     

Impact of chemo-therapy on optimal control of malaria disease with infected immigrants

We derived and analyzed rigorously a mathematical model that describes the dynamics of malaria infection with the recruitment of infected immigrants, treatment of infectives and spray of insecticides against mosquitoes in the population. Both qualitative and quantitative analysis of the deterministic model are performed with respect to stability of the disease free and endemic equilibria. It is found that in the absence of infected immigrants disease-free equilibrium is achievable and is locally asymptotically stable. Using Pontryagin's Maximum Principle, the optimal strategies for disease control are established. Finally, numerical simulations are performed to illustrate the analytical results.     

The effects of mixotrophy on the stability and dynamics of a simple planktonic food web model

Recognition of the microbial loop as an important part of aquatic ecosystems disrupted the notion of simple linear food chains. However, current research suggests that even the microbial loop paradigm is a gross simplification of microbial interactions due to the presence of mixotrophs-organisms that both photosynthesize and graze. We present a simple food web model with four trophic species, three of them arranged in a food chain (nutrients-autotrophs-herbivores) and the fourth as a mixotroph with links to both the nutrients and the autotrophs. This model is used to study the general implications of inclusion of the mixotrophic link in microbial food webs and the specific predictions for a parameterization that describes open ocean mixed layer plankton dynamics. The analysis indicates that the system parameters reside in a region of the parameter space where the dynamics converge to a stable equilibrium rather than displaying periodic or chaotic solutions. However, convergence requires weeks to months, suggesting that the system would never reach equilibrium in the ocean due to alteration of the physical forcing regime. Most importantly, the mixotrophic grazing link seems to stabilize the system in this region of the parameter space, particularly when nutrient recycling feedback loops are included.     

Modeling the impact of early therapy for latent tuberculosis patients and its optimal control analysis

Effective tuberculosis (TB) control depends on case findings to discover infectious cases, investigation of contacts of those with TB, as well as appropriate treatment. Adherence and successful completion of the treatment are equally important. Unfortunately, due to a number of personal, psychosocial, economic, medical, and health service factors, a significant number of TB patients become irregular and default from treatment. In this paper, a mathematical model is developed to assess the impact of early therapy for latent TB and non-adherence on controlling TB transmission dynamics. Equilibrium states of the model are determined and their local stability is examined. With the aid of the center manifold theory, it is established that the model undergoes a backward bifurcation. Qualitative mathematical analysis of the model suggests that a high level of latent tuberculosis case findings, coupled with a decrease of defaulting rate, may be effective in controlling TB transmission dynamics in the community. Population-level effects of organized campaigns to improve early therapy and to guarantee successful completion of each treatment are evaluated through numerical simulations and presented in support of the analytical results.     

Long food chains are in general chaotic

The question whether chaos exists in nature is much debated. In this paper we prove that chaotic parameter regions exist generically in food chains of length greater than three. While nonchaotic dynamics is also possible, the presence of chaotic parameter regions indicates that chaotic dynamics is likely. We show that the chaotic regions survive even at high exponents of closure. Our results have been obtained using a general food chain model that describes a large class of different food chains. The existence of chaos in models of such generality can be deduced from the presence of certain bifurcations of higher codimension.     

Deterministic chaоs and the problem of predictability in population dynamics

The studies of the processes that can significantly influence the predictability in population dynamics are reviewed and the results of mathematical simulations of population dynamics are compared to the time series obtained in field observations. Considerable attention is given to the chaotic changes in population abundance. Some methods of numerical analysis of chaoticity and predictability of the time series are considered. The importance of comparing the results of mathematical simulation and observation data is tightly linked to problems in detecting chaos in the dynamics of natural populations and estimating the prevalence of chaotic regimes in nature. Insight into these problems can allow identification of the functional role of chaotic regimes in population dynamics.     

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.