Stability properties of 2-species models of mutualism: Simulation studies

Summary Most previous analyses of the stability properties of models of mutualism have emphasized the destabilizing effects of mutualism. However, these analyses can be shown to be based upon inappropriate assumptions, or to be applicable only for special cases of mutualism. In this paper three basic 2-species models of mutualism are presented and their six combinations are analyzed by computer simulation for their return time stability and persistence stability. Four out of six models show greater return time stability than an appropriate model without mutualism, and all models show higher persistence stability than the model without mutualism. It is argued that real biological systems can be related to the qualitative structure of each of the basic models of mutualism, and that therefore none of the basic models or their stability properties can be eliminated a priori as being inappropriate. The conclusion follows that while some kinds of mutualistic interactions may be relatively unstable, other mutualisms, probably representing the majority of cases, can be considered to be relatively stable. The limitations of these models and analyses are considered.     

Multi-Platform Compatible Software for Analysis of Polymer Bending Mechanics

Cytoskeletal polymers play a fundamental role in the responses of cells to both external and internal stresses. Quantitative knowledge of the mechanical properties of those polymers is essential for developing predictive models of cell mechanics and mechano-sensing. Linear cytoskeletal polymers, such as actin filaments and microtubules, can grow to cellular length scales at which they behave as semiflexible polymers that undergo thermally-driven shape deformations. Bending deformations are often modeled using the wormlike chain model. A quantitative metric of a polymer''s resistance to bending is the persistence length, the fundamental parameter of that model. A polymer''s bending persistence length is extracted from its shape as visualized using various imaging techniques. However, the analysis methodologies required for determining the persistence length are often not readily within reach of most biological researchers or educators. Motivated by that limitation, we developed user-friendly, multi-platform compatible software to determine the bending persistence length from images of surface-adsorbed or freely fluctuating polymers. Three different types of analysis are available (cosine correlation, end-to-end and bending-mode analyses), allowing for rigorous cross-checking of analysis results. The software is freely available and we provide sample data of adsorbed and fluctuating filaments and expected analysis results for educational and tutorial purposes.     

A universal bifurcation diagram for seasonally perturbed predator-prey models

The bifurcations of a periodically forced predator-prey model (the chemostat model), with a prey feeding on a limiting nutrient, are numerically detected with a continuation technique. Eight bifurcation diagrams are produced (one for each parameter in the model) and shown to be topologically equivalent. These diagrams are also equivalent to those of the most commonly used predator-prey model (the Rosenzweig-McArthur model). Thus, all basic modes of behavior of the two main predator-prey models can be explained by means of a single bifurcation diagram.     

Modelling nutritional mutualisms: challenges and opportunities for data integration

Nutritional mutualisms are ancient, widespread, and profoundly influential in biological communities and ecosystems. Although much is known about these interactions, comprehensive answers to fundamental questions, such as how resource availability and structured interactions influence mutualism persistence, are still lacking. Mathematical modelling of nutritional mutualisms has great potential to facilitate the search for comprehensive answers to these and other fundamental questions by connecting the physiological and genomic underpinnings of mutualisms with ecological and evolutionary processes. In particular, when integrated with empirical data, models enable understanding of underlying mechanisms and generalisation of principles beyond the particulars of a given system. Here, we demonstrate how mathematical models can be integrated with data to address questions of mutualism persistence at four biological scales: cell, individual, population, and community. We highlight select studies where data has been or could be integrated with models to either inform model structure or test model predictions. We also point out opportunities to increase model rigour through tighter integration with data, and describe areas in which data is urgently needed. We focus on plant‐microbe systems, for which a wealth of empirical data is available, but the principles and approaches can be generally applied to any nutritional mutualism.     

Biological control in a disturbed environment

Most ecological and epidemiological models describe systems with continuous uninterrupted interactions between populations. Many systems, though, have ecological disturbances, such as those associated with planting and harvesting of a seasonal crop. In this paper, we introduce host–parasite–hyperparasite systems as models of biological control in a disturbed environment, where the host–parasite interactions are discontinuous. One model is a parasite–hyperparasite system designed to capture the essence of biological control and the other is a host–parasite–hyperparasite system that incorporates many more features of the population dynamics. Two types of discontinuity are included in the models. One corresponds to a pulse of new parasites at harvest and the other reflects the discontinuous presence of the host due to planting and harvesting. Such discontinuities are characteristic of many ecosystems involving parasitism or other interactions with an annual host. The models are tested against data from an experiment investigating the persistent biological control of the fungal plant parasite of lettuce Sclerotinia minor by the fungal hyperparasite Sporidesmium sclerotivorum, over successive crops. Using a combination of mathematical analysis, model fitting and parameter estimation, the factors that contribute the observed persistence of the parasite are examined. Analytical results show that repeated planting and harvesting of the host allows the parasite to persist by maintaining a quantity of host tissue in the system on which the parasite can reproduce. When the host dynamics are not included explicitly in the model, we demonstrate that homogeneous mixing fails to predict the persistence of the parasite population, while incorporating spatial heterogeneity by allowing for heterogeneous mixing prevents fade-out. Including the host''s dynamics lessens the effect of heterogeneous mixing on persistence, though the predicted values for the parasite population are closer to the observed values. An alternative hypothesis for persistence involving a stepped change in rates of infection is also tested and model fitting is used to show that changes in some environmental conditions may contribute to parasite persistence. The importance of disturbances and periodic forcing in models for interacting populations is discussed.     

Using topological equivalence to discover stable control parameters in biodynamic systems

In order to better understand the mechanisms that contribute to low back pain, researchers have developed mathematical models and simulations. A mathematical model including neuromuscular feedback control is developed for a person balancing on an unstable sitting apparatus, the wobble chair. When the application of a direct method fails to discover appropriate controller gain parameters for the wobble chair, we show how topological equivalence can be used to indirectly identify appropriate parameter values. The solution is found by first transforming the wobble chair into the Acrobot, another member of the same family of topologically equivalent dynamical systems. After finding appropriate gain parameters for the Acrobot, a continuous transformation is performed to convert the Acrobot back to the wobble chair, during which the gain parameters are adjusted to maintain stability. Thus, we demonstrate how topological equivalence can be used to indirectly solve a problem that was difficult to solve directly.     

Using topological equivalence to discover stable control parameters in biodynamic systems

In order to better understand the mechanisms that contribute to low back pain, researchers have developed mathematical models and simulations. A mathematical model including neuromuscular feedback control is developed for a person balancing on an unstable sitting apparatus, the wobble chair. When the application of a direct method fails to discover appropriate controller gain parameters for the wobble chair, we show how topological equivalence can be used to indirectly identify appropriate parameter values. The solution is found by first transforming the wobble chair into the Acrobot, another member of the same family of topologically equivalent dynamical systems. After finding appropriate gain parameters for the Acrobot, a continuous transformation is performed to convert the Acrobot back to the wobble chair, during which the gain parameters are adjusted to maintain stability. Thus, we demonstrate how topological equivalence can be used to indirectly solve a problem that was difficult to solve directly.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

Systems analysis of an acarine predator-prey system II: Interactions in discontinuous environment

An acarine predator-prey system in a circular stepping-stone environment was described with a simulation model to elucidate the factors responsible for persistence of the system. The main assumptions in this model are: (1) The prey are inevitably eliminated in patches in which predators exist. (2) The density of prey declines and becomes extinct by plant defoliation due to feeding by prey. In this regard this model is different from the models which mimickedHuffaker's (1958) experiments and assumed stable plant-prey relations. Analyses showed that the critical factor in persistence of the predator-prey system was the plant-prey relations, at any combination of other parameters involved in the model. The predator-prey system did not persist long under the unstable relationship of prey and plant. Otherwise the system persisted longer especially when I used a larger number of patches, a larger amount of plant in each patch, and long-distance-migrations of the prey. In particular, frequent emigration of the prey regardless of plant conditions was most effective.     

A simple persistence condition for structured populations

The fundamental question in both basic and applied population biology of whether a species will increase in numbers is often investigated by finding the population growth rate as the largest eigenvalue of a deterministic matrix model. For a population classified only by age, and not stage or size, a simpler biologically interpretable condition can be used, namely whether R ₀, the mean number of offspring per newborn, is greater than one. However, for the many populations not easily described using only age classes, stage-structured models must be used for which there is currently no quantity like R ₀. We determine analogous quantities that must be greater than one for persistence of a general structured population model that have a similar useful biological interpretation. Our approach can be used immediately to determine the magnitude of changes and interactions that would either allow population persistence or would ensure control of an undesirable species.     

On the degradation of forest ecosystems by extreme events: Statistical Model Checking of a hybrid model

In this paper, we study the vulnerability of forest ecosystems perturbed by extreme events, such as those arising from climate change. To investigate the complex interactions between the biological dynamics of the forest and the climatic activity, we construct an original hybrid model, obtained by coupling a continuous reaction–diffusion system, which describes the spatio-temporal dynamics of the forest ecosystem, with a discrete probabilistic process, which models the possible occurrences of extreme events. Properties of ecological interest are considered: invariance of the persistence equilibrium, attraction to the extinction equilibrium and emergence of degraded states. Those properties of the hybrid model are verified through an extension of the Statistical Model Checking framework. We establish the existence of a threshold above which the persistence equilibrium of the forest ecosystem is compromised and give a numerical assessment of this threshold in terms of the probability and intensity of extreme events. We also present non-trivial parameter conditions for which the forest ecosystem converges to a degraded savanna-like state.     

Prey Selection,Vertical Migrations and the Impacts of Harvesting Upon the Population Dynamics of a Predator-Prey System

A model is developed to describe the interaction between a predator and two prey types located in different regions. Conditions for stability and persistence are analysed. The effects of harvesting the predators are investigated by making the predator mortality rate habitat dependent. Results demonstrate that for any given set of parameter values there is a value of the intrinsic preference of the predator for each prey type at which the system undergoes a Hopf bifurcation. Above this critical value the system evolves towards a stable equilibrium, whereas below it, stable limit cycles arise by Hopf bifurcations. Simulations demonstrate that the presence of demographic stochasticity may destabilise oscillatory populations, thereby causing population extinctions. An application of the model to the foraging behaviour of North Sea cod is described. It is shown that if the preferred prey is more productive, it is likely that the equilibrium will be stable, whereas if the less preferred prey is more productive, populations are likely to display cycles and in the stochastic case become extinct. As cod fishing mortality is increased, the point of bifurcation and region of parameter space for which the system is unstable decreases. An increased understanding of how cod behave may enable fish stocks to be managed more successfully, for example by indicating where marine reserves should be placed.     

Allometric scaling enhances stability in complex food webs

Classic local stability theory predicts that complex ecological networks are unstable and are unlikely to persist despite empiricists' abundant documentation of such complexity in nature. This contradiction has puzzled biologists for decades. While some have explored how stability may be achieved in small modules of a few interacting species, rigorous demonstrations of how large complex and ecologically realistic networks dynamically persist remain scarce and inadequately understood. Here, we help fill this void by combining structural models of complex food webs with nonlinear bioenergetic models of population dynamics parameterized by biological rates that are allometrically scaled to populations' average body masses. Increasing predator–prey body mass ratios increase population persistence up to a saturation level that is reached by invertebrate and ectotherm vertebrate predators when being 10 or 100 times larger than their prey respectively. These values are corroborated by empirical predator–prey body mass ratios from a global data base. Moreover, negative effects of diversity (i.e. species richness) on stability (i.e. population persistence) become neutral or positive relationships at these empirical ratios. These results demonstrate that the predator–prey body mass ratios found in nature may be key to enabling persistence of populations in complex food webs and stabilizing the diversity of natural ecosystems.     

A virtual look at Epstein-Barr virus infection: biological interpretations

The possibility of using computer simulation and mathematical modeling to gain insight into biological and other complex systems is receiving increased attention. However, it is as yet unclear to what extent these techniques will provide useful biological insights or even what the best approach is. Epstein-Barr virus (EBV) provides a good candidate to address these issues. It persistently infects most humans and is associated with several important diseases. In addition, a detailed biological model has been developed that provides an intricate understanding of EBV infection in the naturally infected human host and accounts for most of the virus' diverse and peculiar properties. We have developed an agent-based computer model/simulation (PathSim, Pathogen Simulation) of this biological model. The simulation is performed on a virtual grid that represents the anatomy of the tonsils of the nasopharyngeal cavity (Waldeyer ring) and the peripheral circulation--the sites of EBV infection and persistence. The simulation is presented via a user friendly visual interface and reproduces quantitative and qualitative aspects of acute and persistent EBV infection. The simulation also had predictive power in validation experiments involving certain aspects of viral infection dynamics. Moreover, it allows us to identify switch points in the infection process that direct the disease course towards the end points of persistence, clearance, or death. Lastly, we were able to identify parameter sets that reproduced aspects of EBV-associated diseases. These investigations indicate that such simulations, combined with laboratory and clinical studies and animal models, will provide a powerful approach to investigating and controlling EBV infection, including the design of targeted anti-viral therapies.     

Homochiral Growth Through Enantiomeric Cross-Inhibition

The stability and conservation properties of a recently proposed polymerization model are studied. The achiral (racemic) solution is linearly unstable once the relevant control parameter (here the fidelity of the catalyst) exceeds a critical value. The growth rate is calculated for different fidelity parameters and cross-inhibition rates. A chirality parameter is defined and shown to be conserved by the nonlinear terms of the model. Finally, a truncated version of the model is used to derive a set of two ordinary differential equations and it is argued that these equations are more realistic than those used in earlier models of that form.     

The effect of seasonal host birth rates on population dynamics: the importance of resonance

Many of the simple mathematical models currently in use often fail to capture important biological factors. Here we extend current models of insect-pathogen interactions to include seasonality in the birth rate. In particular, we consider the SIR model with self-regulation when applied to specific cases--rabbit haemorrhagic disease and fox rabies. In this paper, we briefly summarize the results of the model with a constant time-independent birth rate, a, which we then replace with the time dependent birth rate a(t), to investigate how this effects the dynamics of the host population. We can split parameter space into an area in which the model without seasonality has no oscillations, in which case a simple averaging rule predicts the behaviour. Alternatively, in the area where oscillations to the equilibrium do occur in the non-seasonal model, disease persistence is more complicated and we get more complex dynamical behaviour in this case. We apply resonance techniques to discover the structure of the subharmonic modes of the SIR model with self-regulation. We then look at whether many biological systems are likely to display these "resonant" dynamics and find that we would expect them to be widespread.     

Habitat destruction in a simple predator-prey patch model: how predators enhance prey persistence and abundance

We model a metapopulation of predator-prey patches using both spatially implicit or mean-field (MF) and spatially explicit (SE) approaches. We show that in the MF model there are parameter regimes for which prey cannot persist in the absence of predators, but can in their presence. In addition, there are parameter regimes for which prey may persist in isolation, but the presence of predators will increase prey patch density. Predators may thus enhance prey persistence and overall abundance. The key mechanism responsible for this effect is the occurrence of prey dispersal from patches that are occupied by both prey and predators. In addition, these patches should be either long-lived, such as that occurs when predators keep prey from overexploiting its local resource, or the presence of a predator on a patch should significantly enhance the prey dispersal out of that patch. In the SE approach these positive effects of predators on prey persistence and abundance occur for even larger parameter ranges than in the MF model. Prey dispersal from predator-prey patches may thus be important for persistence of both species as a community, independent of the modeling framework studied. Comparison of the MF and SE approaches shows that local dispersal constraints can have the edge over global dispersal for the persistence of the metapopulation in regimes where the two species have a beneficial effect on each other. In general, our model provides an example of feedback in multiple-species metapopulations that can make the implementation of conservation schemes based on single-species arguments very risky.     

Symbolic-numeric estimation of parameters in biochemical models by quantifier elimination

The sequencing of complete genomes allows analyses of the interactions between various biological molecules on a genomic scale, which prompted us to simulate the global behaviors of biological phenomena on the molecular level. One of the basic mathematical problems in the simulation is the parameter optimization in the kinetic model for complex dynamics, and many estimation methods have been designed. We introduce a new approach to estimate the parameters in biological kinetic models by quantifier elimination (QE), in combination with numerical simulation methods. The estimation method was applied to a model for the inhibition kinetics of HIV proteinase with ten parameters and nine variables, and attained the goodness of fit to 300 points of observed data with the same magnitude as that obtained by the previous estimation methods, remarkably by using only one or two points of data. Furthermore, the utilization of QE demonstrated the feasibility of the present method for elucidating the behavior of the parameters and the variables in the analyzed model. Therefore, the present symbolic-numeric method is a powerful approach to reveal the fundamental mechanisms of kinetic models, in addition to being a computational engine.