Geometric criteria for the non-existence of cycles in predator-prey systems with group defense

In this paper, we study the existence of cycles in a predator-prey system in which the prey species is equipped with the group defense capability. Some geometric criteria are developed, relating the location of the two positive equilibria on the prey isocline and the non-existence of cycles. We show that under a general geometric condition, if both positive equilibria lie on a downslope or both lie on an upslope of the prey isocline, cycles do not exist.     

A predator-prey model with infected prey

A predator-prey model with logistic growth in the prey is modified to include an SIS parasitic infection in the prey with infected prey being more vulnerable to predation. Thresholds are identified which determine when the predator population survives and when the disease remains endemic. For some parameter values the greater vulnerability of the infected prey allows the predator population to persist, when it would otherwise become extinct. Also the predation on the more vulnerable prey can cause the disease to die out, when it would remain endemic without the predators.     

Interactions leading to persistence in predator-prey systems with group defence

In previous work (Freedman and Wolkowicz, 1986;Bull. math. Biol. 48, 493–508) it was shown that in a predator-prey system where the prey population exhibits group defence, it is possible that enrichment of the environment could lead to extinction of the predator population. In this paper a third population is introduced and criteria are derived under which persistence of all populations will occur. In particular, criteria for a superpredator and for a competitor to stabilize the system in the sense of persistence are analyzed. Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A 4823. This research was carried out while the author was a visiting scholar at the University of Alberta.     

A model of morphogenetic pattern formation

A model for the morphogenetic movement of surfaces composed of cellular monolayers is proposed. The cells are presumed joined at their lateral surfaces. An otherwise unspecified substance called a "morphogen" is introduced which is the agent of change in the individual cell (or cell-like region). The distribution of these cellular deformations define a surface (the middle surface, through the middle of the cell heights) via equations given for the Gauss and Mean curvatures of the surface defined at each point. The Gauss curvature as a function of the morphogen level determines the metric of the surface "g(u, v)" in conformal co-ordinates u, v. A unique equation for the morphogen distribution over the survace is presented which has the property of size invariance, that is, the model "regulates" without need of further arguments. The two resulting coupled equations for the metric and the morphogen, eqns (4) and (2), both non-linear equations, are to be solved self-consistently, once the individual cell deformation as a function of morphogen is given. The surface geometry determines the morphogen distribution, and the morphogen distribution in turn affects the surface geometry. Extension of the model to two or more morphogens is straightforward, and the key property of "regulation" or size invariance of the model is retained. Numerical integration of the two coupled equations is carried out in the case of axial symmetry, and the results presented by the case that individual cells deform by changing the ratio of their apical to basal areas, as well as their heights. Gastrulation in small regulating holoblastic eggs (e.g. starfish, sea urchin and amphioxus) is discussed in light of the present model.     

Estimating a predator-prey dynamical model with the parameter cascades method

Summary .   Ordinary differential equations (ODEs) are widely used in ecology to describe the dynamical behavior of systems of interacting populations. However, systems of ODEs rarely provide quantitative solutions that are close to real field observations or experimental data because natural systems are subject to environmental and demographic noise and ecologists are often uncertain about the correct parameterization. In this article we introduce "parameter cascades" as an improved method to estimate ODE parameters such that the corresponding ODE solutions fit the real data well. This method is based on the modified penalized smoothing with the penalty defined by ODEs and a generalization of profiled estimation, which leads to fast estimation and good precision for ODE parameters from noisy data. This method is applied to a set of ODEs originally developed to describe an experimental predator–prey system that undergoes oscillatory dynamics. The new parameterization considerably improves the fit of the ODE model to the experimental data sets. At the same time, our method reveals that important structural assumptions that underlie the original ODE model are essentially correct. The mathematical formulations of the two nonlinear interaction terms (functional responses) that link the ODEs in the predator–prey model are validated by estimating the functional responses nonparametrically from the real data. We suggest two major applications of "parameter cascades" to ecological modeling: It can be used to estimate parameters when original data are noisy, missing, or when no reliable priori estimates are available; it can help to validate the structural soundness of the mathematical modeling approach.     

Applications of a model for scale-invariant pattern formation in developing systems

A fundamental problem in developmental biology concerns the proportioning of the developing tissue of a morphallactic system into different cell types in a way that is independent of the overall size of the tissue. The two main models for positional information in pattern formation, the source-sink models and the Turing reaction-diffusion models, have shortcomings that limit their applicability. In a previous paper, we described a model that can produce perfectly scale-invariant spatial patterns and analyzed some of its mathematical properties. In the present paper, we demonstrate some of the shortcomings of the standard reaction-diffusion models and discuss the applicability of our model to developmental systems.     

On a simplified model for pattern formation in honey bee colonies

We present a simplified version of a previously presented model (Camazine et al. (1990)) that generates the characteristic pattern of honey, pollen and brood which develops on combs in honey bee colonies. We demonstrate that the formation of a band of pollen surrounding the brood area is dependent on the assumed form of the honey and pollen removal terms, and that a significant pollen band arises as the parameter controlling the rate of pollen input passes through a bifurcation value. The persistence of the pollen band after a temporary increase in pollen input can be predicted from the model. We also determine conditions on the parameters which ensure the accumulation of honey in the periphery and demonstrate that, although there is an important qualitative difference between the simplified and complete models, an analysis of the simplified version helps us understand many biological aspects of the more complex complete model. Corresponding author     

Travelling waves and pattern formation in a model for fungal development

 Under a variety of conditions, the hyphal density within the expanding outer edge of growing fungal mycelia can be spatially heterogeneous or nearly uniform. We conduct an analysis of a system of reaction-diffusion equations used to model the growth of fungal mycelia and the subsequent development of macroscopic patterns produced by differing hyphal and hence biomass densities. Both local and global results are obtained using analytical and numerical techniques. The emphasis is on qualitative results, including the effects of changes in parameter values on the structure of the solution set. Received 22 November 1995; received in revised form 17 May 1996     

Travelling waves in a tissue interaction model for skin pattern formation

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

Encounters in predator-prey systems: a simple discrete model

Predator-prey systems are often described by exploitation models. These models can mimic experimental data very accurately, but it is sometimes difficult to realize the relationships between the models and the behavior of individual predator and prey animals. A simple discrete model is proposed here that tries to elucidate the connections between: the animals' movements, the predator/prey encounters; and the dynamics in the system as globally represented by the exploitation models. In these models, the term "area of discovery" plays an essential role. This term is shown to be a predictable coefficient that is composed of measurable physical properties of the analyzed predator-prey system. The model takes into account that predators and prey in experimental systems often do not search randomly but prefer some parts of the test area. The model is applied to the mite system Phytoseiulus persimilis/Tetranychus urticae under simple artificial conditions.     

The geometrical analysis of a predator-prey model with two state impulses

Using successor functions and Poincaré-Bendixson theorem of impulsive differential equations, the existence of periodical solutions to a predator-prey model with two state impulses is investigated. By stability theorem of periodic solution to impulsive differential equations, the stability conditions of periodic solutions to the system are given. Some simulations are exerted to prove the results.     

A new model for biological pattern formation

Various non-equilibrium growth models have been used to explore the development of morphology in biological systems. Here we review a class of biological growth models which exhibit fractal structures and discuss the relationship of these models to a variety of other phenomena.     

Three-player social parasitism games: implications for resource defense and group formation

Abstract: Individuals that produce resources are often exploited by several individuals; such exploitation may or may not be tolerated. We modeled the decision of a resource owner to accept one scrounger (the "satellite") and of both of these to accept being joined by another (the "floater"). In general, satellites tolerated floaters when competition between them was low, while owners tolerated satellites when facilitation between satellite and floater was high. When floaters were likely to find resources without joining, owners were more likely to resist satellites. In some cases, Nash equilibria were also mutually beneficial for two of the three individuals. Our model makes the counterintuitive prediction that mutually beneficial coalitions between satellites and floaters can only arise when the net benefits arising from the other's presence are low. When facilitation between satellites and floaters is high, satellites and owners may form mutually beneficial groups and groups with division of labor, or alternatively, owners may benefit from encouraging floaters to join. Finally, our model suggests there must be differences in competitive ability or some benefit of familiarity for owners to tolerate satellites but not floaters. We discuss empirical evidence for these and other predictions of the model.     

Travelling lipid domains in a dynamic model for protein-induced pattern formation in biomembranes

Cell membranes are composed of a mixture of lipids. Many biological processes require the formation of spatial domains in the lipid distribution of the plasma membrane. We have developed a mathematical model that describes the dynamic spatial distribution of acidic lipids in response to the presence of GMC proteins and regulating enzymes. The model encompasses diffusion of lipids and GMC proteins, electrostatic attraction between acidic lipids and GMC proteins as well as the kinetics of membrane attachment/detachment of GMC proteins. If the lipid-protein interaction is strong enough, phase separation occurs in the membrane as a result of free energy minimization and protein/lipid domains are formed. The picture is changed if a constant activity of enzymes is included into the model. We chose the myristoyl-electrostatic switch as a regulatory module. It consists of a protein kinase C that phosphorylates and removes the GMC proteins from the membrane and a phosphatase that dephosphorylates the proteins and enables them to rebind to the membrane. For sufficiently high enzymatic activity, the phase separation is replaced by travelling domains of acidic lipids and proteins. The latter active process is typical for nonequilibrium systems. It allows for a faster restructuring and polarization of the membrane since it acts on a larger length scale than the passive phase separation. The travelling domains can be pinned by spatial gradients in the activity; thus the membrane is able to detect spatial clues and can adapt its polarity dynamically to changes in the environment.