Predator-prey interactions, modeling

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

In biological dissipative structures, self-oiganization may be related to the attractors in the phase space, which correspond to ordered motions of the involved biological elements (De la Fuenta, 1999). When the system is far from equilibrium, ordering in time or spontaneous rhythmic behavior may occur. The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

The Lotka-Volterra model is often used to characterize predator-prey interactions. For example, if R is the population of rabbits (which reproduce autocatalytically), G is the amount of grass available for rabbit food (assumed to be constant), L is the population of lynxes that feeds on the rabbits, and D represents dead lynxes, the following equations represent the dynamic behavior of the populations of rabbits and lynxes ... 

Ca2] R. P. Canale (1970), An analysis of models describing predator-prey interaction, Biotechnology and Bioengineering 12 353-78. 

Such models describe the life history of animals as propagation through the different size or mass classes and need a sophisticated formulations of predator-prey interaction. There are several approaches to describe life histories of copepods by models (Carlotti et al., 2000). A new theoretical formulation to allow the consistent embedding of dynamical copepod models into three-dimensional circulation models was given in Fennel (2001). Examples of simulations for the Baltic were given in Fennel and Neumann (2003). The basic idea is that both biomass and abundance of different stages or mass classes are used as state variables, while the process control is related to mean average individuals in each mass class, that is the ratio of biomass over abundance. 

Davidson, K., Cunningham, A. and Flynn, K.J. (1995) Predator-prey interactions between Isochrysis galbana and Oxyrrhis marine. III. Mathematical modelling of predation and nutrient regeneration. Journal of Plankton Research, 17, 465-492. 

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised Lotka-Volterra equations (De, 1984). The transition probability of extinction is to be determined. The standard procedure is to convert the problem to a Fokker-Planck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. 

This section is meant to contribute to the old problem of the interaction between two biological species. To be or not to be is the essential question decided by the predator-prey interaction for the members of certain species. The famous Volterra-Lotka model for this problem [4.21,22] has attracted many researchers who have tried to generalize it in many respects [4.1, 6, 8, 9, 23-26]. 

The extension of the pure Volterra-Lotka model by adding non-linear migration leads to a complex picture with a relatively large variety of different solution cases. In particular, the result has been obtained that with an appropriate mixture of migration and predator-prey interaction a limit cycle appears which is similar to the original Volterra-Lotka cycles but which is stable and independent of the initial conditions. 

This discussion of a selected set of solutions of the model shows how complex even simple systems including predator-prey interaction and (non-linear) migration can be. This fact may shed a new light on the manifold of channels of survival shown by species in the course of evolution. 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

FW2] H. I. Freedman and P. Waltman (1984), Persistence in a model of three interacting predator-prey populations, Mathematical Bioscience 68 213-31. 

Any effects on populations may ultimately be manifested as effects on communities because, by definition, communities are collections of interacting populations of several species (e.g., an aquatic community may consist of populations of fish, worms, plants, insects). Individual populations within a community may interact by competing for resources (food, habitat, etc.) or by predator/prey relationships. Environmental contaminants can affect the structure of communities as well as the interactions of species within them. For example, it is well known that exposure to chemicals may cause a reduction in community diversity (e.g., relative number of species), and changes in community composition. In addition, the trophic structure of fish and invertebrate communities may also be affected by exposure to anthropogenic chemicals. Changes in community structure and diversity may be determined by field sampling or manipulative studies. Alternatively, computer simulations using food web or linked population models may be used to assess community-level effects. 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

Oscillations can also arise from the nonlinear interactions present in population dynamics (e.g. predator-prey systems). Mixing in this context is relevant for oceanic plankton populations. Phytoplankton-zooplankton (PZ) and other more complicated plankton population models often exhibit oscillatory solutions (see e.g. Edwards and Yool (2000)). Huisman and Weissing (1999) have shown that oscillations and chaotic fluctuations generated by the plankton population dynamics can provide a mechanism for the coexistence of the huge number of plankton species competing for only a few key resources (the plankton paradox ). In this chapter we review theoretical, numerical and experimental work on unsteady (mainly oscillatory) systems in the presence of mixing and stirring. 

Ecological systems can display temporal oscillations in parameter values due to seasonal variations. The effects of time-varying diffusivities on the Turing instability were first considered by Timm and Okubo [436] in a predator-prey model describing the interaction between zooplankton and phytoplankton. Temporal variations in the horizontal diffusion coefficients arise from the interaction of vertical current shear with vertical mixing processes. 

The reaction steps 1.7 and 1.8 are autocatal3 tic in nature, because the X and Y accelerate their own productions. These two reaction processes are also significant for the evolution of the oscillatory cycles. The model was very effective to explain some biological interactions between the ecosystems. This model is also known as predator-prey theory. 

Far from equilibrium, we only keep the forward reactions. Using the linear stability theory, show that the perturbations around the nonequilibrium steady state lead to oscillations in [X] and [Y], as was discussed in section 18.2. This model was used by Lotka and Volterra to describe the struggle of life (see V. Volterra, Theorie mathematique de la Lutte pour la Vie, Paris Gauthier Villars, 1931). Here X is the prey (lamb) and Y is the predator (wolf). This model of the prey-predator interaction shows that the populations X and Y will exhibit oscillations. 

First model for oscillating system was proposed by Volterra for prey-predator interactions in biological systems and by Lotka for autocatalytic chemical reactions. Lotka s model can be represented as... 

As indicated above, theoretical models for biological rhythms were first used in ecology to study the oscillations resulting from interactions between populations of predators and preys [6]. Neural rhythms represent another field where such models were used at an early stage The formalism developed by Hodgkin and Huxley [7] stiU forms the core of most models for oscillations of the membrane potential in nerve and cardiac cells [33-35]. Models were subsequently proposed for oscillations that arise at the cellular level from regulation of enzyme, receptor, or gene activity (see Ref. 31 for a detailed fist of references). 

When a more detailed analysis of microbial systems is undertaken, the limitations of unstructured models become increasingly apparent. The most common area of failure is that where the growth is not exponential as, for example, during the so-called lag phase of a batch culture. Mathematically, the analysis is similar to that of the interaction of predator and prey, involving a material balance for each component being considered. 

In many recent studies, the role of terpenoids in indirect defense has been studied. A broad range of plant species such as Arabidopsis, com, lima bean, cucumber, tomato, tobacco, apple, and poplar serves as models for studies on the genetic, biochemical, physiological, and ecological aspects of these tritrophic interactions between plants, herbivores, and natural enemies (e.g.. Reference 10). Plants have been shown to respond with quantitatively and qualitatively different volatile blends to different herbivore species, and predators can exploit this behavior to respond specifically to their prey (Fig. 3)... 

The model describes a three level vertical food-chain, where the resource or vegetation u is consumed by herbivores u, which in turn are prey on by top predators w. The parameters a, b and c represent the respective growth rates of each species in the absence of interspecific interactions (fci = fc2 = 0). The functions fi x,y) describe interactions between... 

The model equations for simple prey-predator interactions, initially according to Lotka and Volterra and later refined by Bungay and Bungay (1968), are as follows ... 

The model which we shall discuss in the present section is concerned with an ecological problem. Almost 50 years ago, VOLTERRA (1928, 1931) set up a system of differential equations for the interaction of a prey species X and a predator species Y ... 

Mixed Populations. Considerable literature has evolved on mixed populations kinetics. Virtually every possible biological interaction from the classical prey-predator model of Lotka (35) to various forms of commensalism, synergism, etc. have been modelled. Virtually all models end up exhibiting stable oscillations. Solutions are often expressed in triangular phase plane plots of the limiting substrate and the two species. Invariably the plots have a limit cycle as one of the stable steady state solutions. From this one gains the impression that oscillatory behavior is the norm rather than the exception in biological reactors. 

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