Coexistence of species in a defensive switching model

We consider a simple mathematical model of two-predators and one-prey system which has the defensive switching property of predation-avoidance. We assume that the prey remains vigilant against relatively abundant predator species and guards against it by switching to another (relatively rare) predator species. We analyze how the intensity of defensive switching affects the stability of the model system. It is seen that the system generally has a stable three species coexisting equilibrium state. In the special case that the intensity of defensive switching equals one and the two predators have the same mortality rates, it is shown that the system asymptotically settles to a Volterra's oscillation in three-dimensional space. It is observed that a sufficiently small or sufficiently large value of intensity of defensive switching can make the system unstable. Finally, it is shown that the handling time may have a stabilizing effect on predator-prey systems with defensive switching.     

Global stability in two species interactions

Summary It is proved that sufficient conditions for global stability in a Lotka-Volterra model of a two species interactions are (i) the equilibrium is feasible, (ii) the equilibrium is locally stable and (iii) at the equilibrium each species sustains density dependent mortalities due to intraspecific interactions.     

Switching effect of predation on competitive prey species

The fact that the predation pressure has a stabilizing effect on the community of competitive species is demonstrated by a mathematical model of two-preys and one-predator system which has the switching property of predation. By analyzing a dynamical system for these three species populations, it is shown that, in a wide range of parameter space, the system has stable coexisting equilibrium states and the manifold of stable stationary points exhibits a cusp catastrophe and there exist two stable stationary points in the cusp region in the parameter space. Thus, it has been shown that Cause's competitive exclusion is actually relaxed by the switching mechanism of predation.     

Direct demonstration that homotetrameric chaperone SecB undergoes a dynamic dimer-tetramer equilibrium

We have shown here that the cytosolic bacterial chaperone SecB is a structural dimer of dimers that undergoes a dynamic equilibrium between dimer and tetramer in the native state. We demonstrated this equilibrium by mixing two tetrameric species of SecB that can be distinguished by size. We showed that the homotetrameric species exchanged dimers, because when the mixture was analyzed both by size exclusion chromatography and native polyacrylamide gel electrophoresis a third hybrid tetrameric species was detected. Furthermore, treatment of SecB with 5,5'-dithiobis-(2-nitrobenzoic acid), which modifies the sulfhydryl group on cysteines, caused irreversible dissociation to a dimer indicating that cysteine must be involved in the stabilizing interactions at the dimer interface. It is clear that the two dimer-dimer interfaces of the SecB tetramer are differentially stable. Dissociation at one interface allows for a dynamic dimer-tetramer equilibrium. Because only dimers were exchanged it is clear that the other interface between dimers is significantly more stable, otherwise oligomers should have formed with a random distribution of monomers.     

Two fish species competition model with nonlinear interactions and equilibrium catches

Two species competition model is built up by assuming the hypothetical second order interactions in order to consider effects of exploitation on two competing fish species with non-linear interactions. Most important characteristic of this model, compared withLotka-Volterra type linear competition model, is that this model can possess multiple stable equilibrium points. Therefore there is a possibility that two species keeping the equilibrium state at one stable equilibrium point will be attracted to the other stable equilibrium point after a heavy perturbation. In this model reversible change of the fishing pressure does not always results in that of the equilibrium catch. In this sence MSY concept for single species can not be extended to this model. If there are multiple stable equilibrium points, the change of the dominant fish species, catastrophic and irreversible change of each equilibrium catch may be observed when the perturbation by the exploitation is added. This phenomenon immediately reminds us of the change of the dominant fish species between Japanese common mackerel and Pacific saury in the northwest Pacific Ocean. In case of the management of two competing fish species with nonlinear interactions, the consideration on the balance between the fishing pressure for each species may be as important as the decision on the catch limit for each species. MSY level for each species based on the single-species theory could be quite erroneous.     

Moments of the probability distribution of moving molecules undergoing reversible isomerization,with applications to experiments

Exact expressions are obtained for the mean position and the variance about the mean of macromolecules which are moving in an electrostatic or centrifugal field and which, at the same time, are switching back and forth between two isomeric states. Comparison with experiment then yields the forward and backward switching rates. The following special cases are considered: (a) only one species present initially; (b) both species present initially but not in their equilibrium proportions; (c) both species present initially in their equilibrium proportions. It is shown that in the first two cases we need only measure the mean position of all the molecules in order to measure the absolute switching rates k₁ and k₂. In the third case, however, we must measure the variance (mean-square deviation) of the position in order to obtain k₁ and k₂. The first two situations arise when “jumps” (e.g., in temperature or pressure) are made, while the third situation is obtained if the experiment is conducted with the species in chemical equilibrium throughout the experiment.     

Effects of convective and dispersive interactions on the stability of two species

The evolution and local stability of a system of two interacting species in a finite two-dimensional habitat is investigated by taking into account the effects of self- and cross-dispersion and convection of the species. In absence of cross-dispersion, an equilibrium state which is stable without dispersion is always stable with dispersion provided that the dispersion coefficients of the two species are equal. However, when the dispersion coefficients of the two species are different, the possibility of self-dispersive instability arises. It is also pointed out that the cross-dispersion of species may lead to stability or instability depending upon the nature and the magnitude of the cross-dispersive interactions in comparison to the self-dispersive interactions. The self-convective movement of species increases the stability of the equilibrium state and can stabilize an otherwise unstable equilibrium state. The effect of cross-convection (in absence of self-dispersion and self-convection) is to stabilize the equilibrium state in a prey-predator model with positive cross-dispersion coefficients for the prey species. Finally, it is shown that if the system is stable under homogeneous boundary conditions it remains so under non-homogeneous boundary conditions.     

Population dynamical consequences of reduced predator switching at low total prey densities

Several models of rapid switching by a predator in a two-prey environment are analyzed. The goal is to determine how the dynamics of the system and the potential indirect effects between prey are affected by the dependence of switching on total prey density. In exploring this question, the difference between the population-level consequences of switching in stable and cycling predator-prey systems is also examined. We concentrate on reduced switching at low densities, a feature that is likely because of the difficulty of distinguishing between two very low densities. The main findings are: (1) switching in unstable systems can produce positive indirect effects of one prey species on the other; and (2) reduced switching at low densities can greatly alter the dynamics of the system and the indirect effects between prey. Both of the possibilities are only evident in cycling systems. Reduced switching at low total prey densities leads to heavier predation on the slower-growing prey when both prey species are rare. As a consequence, there is a lag in the recovery of the slower-growing prey species after predator densities fall, and the dynamics of the two prey become desynchronized. The net result is increased indirect interactions between prey, and a greater likelihood of exclusion of the slower growing prey. The analysis of these models suggests a need for more empirical work to determine whether switching is reduced by very low total prey densities, and to study the long-term dynamics that occur in systems with switching predators.     

Effects of dispersal on the stability of a gonorrhea endemic model

Using Liapunov's direct method, effects of dispersal on the linear and nonlinear stability of the endemic equilibrium state of the system governing the spread of gonorrhea are investigated. It is noted that the equilibrium state, which is nonlinearly asymptotically stable in the feasible region of the phase plane in the absence of dispersal, remains so with self-dispersal also (cross-dispersal being absent). However, in the presence of both self- and cross-dispersal, the equilibrium state can still remain nonlinearly asymptotically stable in the entire feasible region provided a certain condition involving self- and cross-dispersal coefficients is satisfied. It is also seen in this case that, for the linearly stable equilibrium state, there exists a subregion of the feasible region where it is nonlinearly asymptotically stable.     

The dynamics of two-species allelopathic competition with optimal harvesting

This paper analyses a bionomic model of two competitive species in the presence of toxicity with different harvesting efforts. An interesting dynamics in the first quadrant is analysed and two saddle-node bifurcations are detected for different bifurcation parameters. It is noted that under certain parametric restrictions, the model has a unique positive equilibrium point that is globally asymptotically stable whenever it is locally stable. It is also noted that the model can have zero, one or two feasible equilibria appearing through saddle-node bifurcations. The non-existence of a limit cycle in the interior of the first quadrant is also discussed using the Poincare–Dulac criteria. The saddle-node bifurcations are studied using Sotomayor's theorem. Numerical simulations are carried out to validate the analytical findings. The conditions for the existence of bionomic equilibria are discussed and an optimal harvesting policy is derived using Pontryagin's maximum principle.     

The dynamics of two-species allelopathic competition with optimal harvesting

This paper analyses a bionomic model of two competitive species in the presence of toxicity with different harvesting efforts. An interesting dynamics in the first quadrant is analysed and two saddle-node bifurcations are detected for different bifurcation parameters. It is noted that under certain parametric restrictions, the model has a unique positive equilibrium point that is globally asymptotically stable whenever it is locally stable. It is also noted that the model can have zero, one or two feasible equilibria appearing through saddle-node bifurcations. The non-existence of a limit cycle in the interior of the first quadrant is also discussed using the Poincare-Dulac criteria. The saddle-node bifurcations are studied using Sotomayor's theorem. Numerical simulations are carried out to validate the analytical findings. The conditions for the existence of bionomic equilibria are discussed and an optimal harvesting policy is derived using Pontryagin's maximum principle.     

Predator mediated coexistence with a switching predator

Predator mediated coexistence of two competing species with general frequency dependent switching in the predator is examined. The stability criterion used is “permanent coexistence.” This is a global criterion which ensures that eventually the species end up in a region M of phase space separated from the boundary (corresponding to extinction of at least one of the species), but which places no restriction on the behavior in M, and so allows, for example, the existence of a stable limit cycle. The principal determinant of survival turns out to be the strength of the switching when one prey is rare, the form of the switching elsewhere being irrelevant. It is found that strong switching is a powerful influence for coexistence. However, in contrast with the conclusion of previous investigations, we find that the influence of switching is complex, and that under some circumstances weak switching can actually destroy coexistence in a system which without switching leads to survival of all species.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

Multilocus genetics and the coevolution of quantitative traits

We develop and analyze an explicit multilocus genetic model of coevolution. We assume that interactions between two species (mutualists, competitors, or victim and exploiter) are mediated by a pair of additive quantitative traits that are also subject to direct stabilizing selection toward intermediate optima. Using a weak-selection approximation, we derive analytical results for a symmetric case with equal locus effects and no mutation, and we complement these results by numerical simulations of more general cases. We show that mutualistic and competitive interactions always result in coevolution toward a stable equilibrium with no more than one polymorphic locus per species. Victim-exploiter interactions can lead to different dynamic regimes including evolution toward stable equilibria, cycles, and chaos. At equilibrium, the victim is often characterized by a very large genetic variance, whereas the exploiter is polymorphic in no more than one locus. Compared to related one-locus or quantitative genetic models, the multilocus model exhibits two major new properties. First, the equilibrium structure is considerably more complex. We derive detailed conditions for the existence and stability of various classes of equilibria and demonstrate the possibility of multiple simultaneously stable states. Second, the genetic variances change dynamically, which in turn significantly affects the dynamics of the mean trait values. In particular, the dynamics tend to be destabilized by an increase in the number of loci.     

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

Competition and stoichiometry: coexistence of two predators on one prey

The competitive exclusion principle (CEP) states that no equilibrium is possible if n species exploit fewer than n resources. This principle does not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can contribute to the stable maintenance of biodiverse communities. Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements such as carbon (C), nitrogen (N), and phosphorus (P) that are present in various proportions in organisms. We incorporate these principles into a standard predator-prey model to analyze competition between two predators on one autotrophic prey. The model tracks two essential elements, C and P, in each species. We show that a stable equilibrium is possible with two predators on this single prey. At this equilibrium both predators can be limited by the P content of the prey. The analysis suggests that chemical heterogeneity within and among species provides new mechanisms that can support species coexistence and that may be important in maintaining biodiversity.     

Classical and resource-based competition: a unifying graphical approach

A graphical technique is given for determining the outcome of two species competition for two resources. This method is unifying in the sense that the graphical criterion leading to the various outcomes of competition are consistent across most of the spectrum of resource types (from those that fulfill the same growth needs to those that fulfill different needs) regardless of the classification method used, and the resulting graphs bear a striking resemblance to the well-known phase portraits for two species Lotka–Volterra competition. Our graphical method complements that of Tilman. Both include zero net growth isoclines. However, instead of using the consumption vectors at potential coexistence equilibria to determine input resource concentrations leading to specific competitive outcomes, we introduce curves bounding the feasible set (the set where the resource concentrations of any equilibrium solution must be located). The washout equilibrium (corresponding to the supply point) occurs at an intersection of curves defining the feasible set boundary. The resource concentrations of all other equilibria are found where zero net growth isoclines either intersect each other inside the feasible set or they intersect the feasible set boundary. A species has positive biomass at such an equilibrium only if its zero net growth isocline is involved in such an intersection. The competitive outcomes are then determined from the position of the single species equilibria, just as in the phase portrait analysis for classical competition (rather than from information at potential coexistence equilibria as in Tilman’s method).     

Association between competition and obligate mutualism in a chemostat

In this paper, we consider a simple chemostat model involving two obligate mutualistic species feeding on a limiting substrate. Systems of differential equations are proposed as models of this association. A detailed qualitative analysis is carried out. We show the existence of a domain of coexistence, which is a set of initial conditions in which both species survive. We demonstrate, under certain supplementary assumptions, the uniqueness of the stable equilibrium point which corresponds to the coexistence of the two species.     

Association between competition and obligate mutualism in a chemostat

In this paper, we consider a simple chemostat model involving two obligate mutualistic species feeding on a limiting substrate. Systems of differential equations are proposed as models of this association. A detailed qualitative analysis is carried out. We show the existence of a domain of coexistence, which is a set of initial conditions in which both species survive. We demonstrate, under certain supplementary assumptions, the uniqueness of the stable equilibrium point which corresponds to the coexistence of the two species.