Global stability of predator-prey interactions

Summary A Lyapunov function is given that extends functions used by Volterra, Goh, and Hsu to a wide class of predator-prey models, including Leslie type models, and a biological interpretation of this function is given. It yields a simple stability criterion, which is used to examine the effect on stability of intraspecific competition among both prey and predators, of a refuge for the prey, and of Holling type II and type III functional responses. Although local stability analysis of these specific models has been done previously, the Lyapunov function facilitates study of global stability and domains of attraction and provides a unified theory which depends on the general nature of the interactions and not on the specific functions used to model them.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

A three dimensional nutrient-plant-herbivore model was proposed and conditions for boundedness, positive invariance, existence and stability of different equilibrium points, Hopf-bifurcation and global stability were obtained. We performed numerical simulations to observe the simultaneous effect of the top-down and the bottom-up mechanism on the system. It was found that nutrient enrichment destroyed the coexistence steady state of the system. This nutrient enrichment could be due to high nutrient input rate or high nutrient recycling rate. In both cases the system showed instability. Moreover, these results were independent of the grazing pressure and the predation functional form.     

Mathematical models of microbial growth and competition in the chemostat regulated by cell-bound extracellular enzymes

A mathematical model of growth and competitive interaction of microorganisms in the chemostat is analyzed. The growth-limiting nutrient is not in a form that can be directly assimilated by the microorganisms, and must first be transformed into an intermediate product by cell-bound extracellular enzymes. General monotone functions, including Michaelis-Menten and sigmoidal response functions, are used to describe nutrient conversion and growth due to consumption of the intermediate product. It is shown that the initial concentration of the species is an important determining factor for survival or washout. When there are two species whose growth is limited by the same nutrient, three different modes of competition are described. Competitive coexistence steady states are shown to be possible in two of them, but they are always unstable. In all of our numerical simulations, the system approaches a steady state corresponding to the washout of one or both of the species from the chemostat.Research supported by NSF grant DMS-90-96279Research supported by NSERC grant A-9358     

Linking saturation,stability and sustainability in food webs with observed equilibrium structure

Stability of a dynamic equilibrium in a predator-prey system depends both on the type of functional response and on the point of equilibrium on the response curve. Saturation effects from Holling type II responses are known to destabilise prey populations, while a type III (sigmoid) response curve has been shown to provide stability at lower levels of saturation. These effects have also been shown in multi-trophic model systems. However, stability analyses of observed equilibria in real complex ecosystems have as yet not assumed non-linear functional responses. Here, we evaluate the implications of saturation in observed balanced material-flow structures, for system stability and sustainability. We first make the effects of the non-linear functional responses on the interaction strengths in a food web transparent by expressing the elements of Jacobian ‘community’ matrices for type II and III systems as simple functions of their linear (type I) counterparts. We then determine the stability of the systems and distinguish two critical saturation levels: (1) a level where the system is just as stable as a type I system and (2) a level above which the system cannot be stable unless it is subsidised, separating a stable materially sustainable regime from an unsustainable one. We explain the stabilising and destabilising effects in terms of the feedbacks in the systems. The results shed light on the robustness of observed patterns of interaction strengths in complex food webs and suggest the implausibility of saturation playing a significant role in the equilibrium dynamics of sustainable ecosystems.     

Qualitative dynamics of three species predator-prey systems

Summary A qualitative analysis of some two and three species predator-prey models is achieved by application of the method of averaging in conjunction with a Lyapunov function constructed from the appropriate Volterra-Lotka model. We calculate the limit cycle solution for a two-species model with a Holling type functional response of the predator to its prey by means of a time-scaled transformation. The existence of a bifurcation of steady states for a community of three species is discussed and the periodic solution around one of the unstable steady states is calculated to the lowest approximation. Several comments are made regarding the behavior of these systems under changes of some control parameters.This work was supported in parts by USERDA, Contract number E(11-1)-3001.     

Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat

Recent research indicates that viruses are much more prevalent in aquatic environments than previously imagined. We derive a model of competition between two populations of bacteria for a single limiting nutrient in a chemostat where a virus is present. It is assumed that the virus can only infect one of the populations, the population that would be a more efficient consumer of the resource in a virus free environment, in order to determine whether introduction of a virus can result in coexistence of the competing populations. We also analyze the subsystem that results when the resistant competitor is absent. The model takes the form of an SIS epidemic model. Criteria for the global stability of the disease free and endemic steady states are obtained for both the subsystem as well as for the full competition model. However, for certain parameter ranges, bi-stability, and/or multiple periodic orbits is possible and both disease induced oscillations and competition induced oscillations are possible. It is proved that persistence of the vulnerable and resistant populations can occur, but only when the disease is endemic in the population. It is also shown that it is possible to have multiple attracting endemic steady states, oscillatory behavior involving Hopf, saddle-node, and homoclinic bifurcations, and a hysteresis effect. An explicit expression for the basic reproduction number for the epidemic is given in terms of biologically meaningful parameters. Mathematical tools that are used include Lyapunov functions, persistence theory, and bifurcation analysis.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.     

Operability of Continuous Bioprocesses: Static Behavior of a Large Class of Unstructured Models

The complete static behavior of a large class of unstructured models of continuous bioprocesses is classified using elementary concepts of the singularity theory and continuation techniques. The class consists of models for which the cell growth rate is proportional to the rate of utilization of limiting substrate while the kinetics of cell growth, utilization of limiting substrate and synthesis of the desired non-biomass product are allowed to assume general forms of substrate and product. This class of models was used extensively in the literature to model fermentation processes. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of wash-out conditions with non-trivial steady states and multistability resulting from hysteresis. These general results are applied to a number of experimentally validated models of fermentation processes, and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models. Practical criteria are also derived for the safe operation of the bioprocesses.     

Persistence of pollination mutualisms in plant-pollinator-robber systems

Interactions between pollinators, nectar robbers, defensive plants and non-defensive plants are characterized by evolutionary games, where payoffs for the four species are represented by population densities at steady states in the corresponding dynamical systems. The plant-robber system is described by a predator-prey model with the Holling II functional response, while the plant-pollinator system is described by a cooperative model with the Beddington-DeAngelis functional response. By combining dynamics of the models with properties of the evolutionary games, we show mechanisms by which pollination mutualisms could persist in the presence of nectar robbers. The analysis leads to an explanation for persistence of plant-pollinator-robber systems in real situations.     

Stability Analysis of a Model of Interaction Between the Immune System and Cancer Cells in Chronic Myelogenous Leukemia

We describe here a simple model for the interaction between leukemic cells and the autologous immune response in chronic phase chronic myelogenous leukemia (CML). This model is a simplified version of the model we proposed in Clapp et al. (Cancer Res 75:4053–4062, 2015). Our simplification is based on the observation that certain key characteristics of the dynamics of CML can be captured with a three-compartment model: two for the leukemic cells (stem cells and mature cells) and one for the immune response. We characterize the existence of steady states and their stability for generic forms of immunosuppressive effects of leukemic cells. We provide a complete co-dimension one bifurcation analysis. Our results show how clinical response to tyrosine kinase inhibitors treatment is compatible with the existence of a stable low disease, treatment-free steady state.     

Time-delayed model of RNA interference

RNA interference (RNAi) is a fundamental cellular process that inhibits gene expression through cleavage and destruction of target mRNA. It is responsible for a number of important intracellular functions, from being the first line of immune defence against pathogens to regulating development and morphogenesis. In this paper we consider a mathematical model of RNAi with particular emphasis on time delays associated with two aspects of primed amplification: binding of siRNA to aberrant RNA, and binding of siRNA to mRNA, both of which result in the expanded production of dsRNA responsible for RNA silencing. Analytical and numerical stability analyses are performed to identify regions of stability of different steady states and to determine conditions on parameters that lead to instability. Our results suggest that while the original model without time delays exhibits a bi-stability due to the presence of a hysteresis loop, under the influence of time delays, one of the two steady states with the high (default) or small (silenced) concentration of mRNA can actually lose its stability via a Hopf bifurcation. This leads to the co-existence of a stable steady state and a stable periodic orbit, which has a profound effect on the dynamics of the system.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Lyapunov functions for a generalized Gause-type model

Lyapunov functions are given to prove the global asymptotic stability of a large class of predator-prey models, including the ones in which the intrinsic growth rate of the prey follows the Ricker-law or the Odell generalization of the logistic law, and the functional predator response is of Holling type.Work supported by M.U.R.S.T., 60%.     

The stability of ecosystems: A brief overview of the paradox of enrichment

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system,thereby collapsing the trophic interaction,a phenomenon referred to as "the paradox of enrichment". After it was first pro posed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator -prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form,inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that,although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms,which may differ among systems,need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models.     

Thermal acclimation modulates the impacts of temperature and enrichment on trophic interaction strengths and population dynamics

Global change affects individual phenotypes and biotic interactions, which can have cascading effects up to the ecosystem level. However, the role of environmentally induced phenotypic plasticity in species interactions is poorly understood, leaving a substantial gap in our knowledge of the impacts of global change on ecosystems. Using a cladoceran–dragonfly system, we experimentally investigated the effects of thermal acclimation, acute temperature change and enrichment on predator functional response and metabolic rate. Using our experimental data, we next parameterized a population dynamics model to determine the consequences of these effects on trophic interaction strength and food‐chain stability. We found that (1) predation and metabolic rates of the dragonfly larvae increase with acute warming, (2) warm‐acclimated larvae have a higher maximum predation rate than cold‐acclimated ones, and (3) long‐term interaction strength increases with enrichment but decreases with both acclimation and acute temperatures. Overall, our experimental results show that thermal acclimation can buffer negative impacts of environmental change on predators and increase food‐web stability and persistence. We conclude that the effect of acclimation and, more generally, phenotypic plasticity on trophic interactions should not be overlooked if we aim to understand the effects of climate change and enrichment on species interaction strength and food‐web stability.     

Effects of macromolecular crowding on the structural stability of human α-lactalbumin

The folding of protein, an important process for protein to fulfill normal functions, takes place in crowded physiological environments. α-Lactalbumin, as a model system for protein-folding studies, has been used extensively because it can form stable molten globule states under a range of conditions. Here we report that the crowding agents Ficoll 70, dextran 70, and polyethylene glycol (PEG) 2000 have different effects on the structural stability of human α-lactalbumin (HLA) represented by the transition to a molten globule state: dextran 70 dramatically enhances the thermal stability of Ca(2+)-depleted HLA (apo-HLA) and Ficoll 70 enhances the thermal stability of apo-HLA to some extent, while PEG 2000 significantly decreases the thermal stability of apo-HLA. Ficoll 70 and dextran 70 have no obvious effects on trypsin degradation of apo-HLA but PEG 2000 accelerates apo-HLA degradation by trypsin and destabilizes the native conformation of apo-HLA. Furthermore, no interaction is observed between apo-HLA and Ficoll 70 or dextran 70, but a weak, non-specific interaction between the apo form of the protein and PEG 2000 is detected, and such a weak, non-specific interaction could overcome the excluded-volume effect of PEG 2000. Our data are consistent with the results of protein stability studies in cells and suggest that stabilizing excluded-volume effects of crowding agents can be ameliorated by non-specific interactions between proteins and crowders.     

Ontogenetic shift in <Emphasis Type="Italic">Daphnia</Emphasis>-algae interaction strength altered by stressors: revisiting Jensen’s inequality

Interaction strength among species plays a crucial role in shaping the functioning of ecological communities, but it is often assumed to be insensitive to inter-individual variation in underlying parameters such as attack rates or handling time. Ecological factors including stressors exert age/size-dependent effects on such behavioral parameters, promoting shifts in the distribution of parameter values over ages. Here we analyze the effects of the pesticide methamidophos on the Daphnia-microalga interaction strength. We first analyze age-dependent effects of the pesticide on the Daphnia functional response, and then decompose the population-level effects of the stressor into contributions of shifts in elevation (i.e., vertical effect) versus shifts in nonlinearity (i.e., nonlinear effect) of the response of interaction strength over consumer age. Our results show that (1) Rogers and Holling type II functional response models best fitted the empirical functional responses of Daphnia of different ages, (2) attack rate and handling time were affected by the pesticide, (3) these effects were age-specific, shifting the average attack rate and both the mean and coefficient of variation of handling time of different age classes, (4) population level interaction strength was affected by pesticide exposure by variation in both elevation and nonlinearity of its response over consumer age. We show that both vertical and nonlinear effects were important in magnitude but opposite in sign. The consequences of factors that exert age/size dependent effects can only be evaluated through properly considering inter-individual variation.     

Mathematical analysis of an activation-inhibition system in a discrete field

The activation-inhibition model of Meinhardt and Gierer is investigated in the particular case of a stringlike set of cells. This reaction-diffusion system is considered, from a topical point of view, as a nonautonomous dynamic system which rules the behavior of each cell. It was shown that, when no diffusion occurs, this system can have either no or one or two stable steady states. The effect of intercellular exchanges on the existence and stability of these steady states is studied, so as to apply this model to the investigation of the branching of a filamentous lower plant.