Predator-prey dynamics

The term f(P)Z in (3.71), when f P) is given by (3.74), leads formally to the Michaelis-Menten dynamics (3.39), if Et is identified with the predator density and P with the substrate. This analogy has been elaborated in the literature. For example Real (1977) describes predator-prey dynamics with the Michaelis-Menten scheme (3.27), with S the prey, C the intermediate state of the prey when it is eaten, E is the predator searching for food and P is the new predator biomass produced during the consumption process, so that Et = E + P is the total amount of predator. This leads to a justification... 

In the absence of coupling each autonomous oscillator, e R", follows its own local (predator-prey) dynamics x = F(xi, Xi) which we assume to be either a limit cycle or phase coherent chaos. The oscillators are coupled by local dispersal with strength e over a predefined set Ni of next neighbours and using the diagonal coupling matrix C = diag(ci, C2. .., c ). 

Quite clearly, the growth of a predator population is in some way dependent upon the abundance of its prey. The most frequently cited model of predator-prey dynamics is the set of linked, non-linear differential equations known as the Lotka-Volterra equations (1 ). This model assumes that in the absence of predator, the prey grows exponentially, while in the absence of prey the predator dies exponentially, and that the predator growth rate is directly proportional to the product of the prey... 

The fact that bistability is found in such disparate systems as autocataly-tic chemical reaction kinetics and predator-prey dynamics such as that associated with the spruce budworm has led to the concept of normal forms, dynamical models that illustrate the phenomenon in question and are the simplest possible expression of this phenomenon. Physically meaningful equations, such as the reaction rate law for the iodate-arsenite system described above, can, in principle, always be reduced to the associated normal form. Adopting the usual notation of an overhead dot for time differentiation, the normal form for bistability is the following... 

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. 

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. 

Beddington, J.R., Free, C.A. and Lawton, J.H. (1975). Dynamic complexity in predator-prey models framed in difference equations, Nature 255, 58-60. 

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 ... 

Metapopulation models have been used to examine the dynamics of populations resulting from pesticide application. Sherratt and Jepson (1993) have investigated the impacts of pesticides to invertebrates using single-species and a predator-prey metapopulation models. In the case of the polyphagous predator, persistence of the population in the landscape is enhanced if only a few fields are sprayed, the application rate of the pesticide is low, or the intrinsic toxicity of the pesticide is low. There also appears to be an optimal dispersal rate that maximizes the likelihood of persistence of the predator in a sprayed field. Importantly, there are also patterns of pesticide application that would cause the prey insect population to reach higher densities than would occur otherwise. Dispersal rates of the predator and the prey are important factors determining the prey population densities. The importance of dispersal in the determination of the persistence of a population in a contaminated landscape was discovered in a subsequent study. 

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. 

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. 

To overcome these shortcomings, modified predator-prey models were sought for which should improve the most important oversimplifications of the Lotka-Volterra model. In general, the dynamics of two populations, which are coupled by predation, can be generalized as follows... 

The major new property, when including a type-III food uptake in the predator prey model (15.5) is that the prey isocline becomes a cubic-like function. Again, the dynamics depend on the exact location of the intersections of predator- and prey-isoclines. If the intersection is in one of the two decreasing branches of the prey isocline (see Fig. 15.7), the model exhibits a stable fixed point. Otherwise, the fixed point becomes unstable giving rise to limit cycle oscillations (see Fig. 15.6). 

Many population cycles have the unusual property that their period length remains remarkably constant while their abundance levels are highly erratic. Fig. 15.8a demonstrates these features for one of the most celebrated time series in Ecology - the Canadian hare-lynx cycle. In [28, 29] is was shown that such more complex oscillations can be achieved in simple predator-prey models by including more trophic levels. To describe the main dynamics of the lynx the following ecological foodweb model was presented... 

As demonstrated in Fig. 15.13, these concentric target waves do not result from the chaotic dynamics, but are present in a very similar form also in disordered limit-cycle predator-prey models. This observation suggests that the origin of the target waves may be found in the intrinsic heterogeneity of the considered spatial models, which will be discussed in detail below. 

Some biological responses are periodic, varying in a predictable manner between two limits. The firing of certain nerve cells and the levels of circulating hormones are two examples of periodic responses. Another example is predator-prey population dynamics (see also Section 6.20.3). Oscillatory behavior (Figure 4.3.2) is described mathematically by combinations of sines and... 

Murdie, G. and Hassel, M. P. (1973) Food distribution, searching success and predator-prey models. In The Mathematical Theory of the Dynamics of Biological Populations (Hiorns, R. W., ed.) pp. 87-101. Academic Press, New York. 

Dynamic behavior and oscillations are also found in nature, such as predator-prey interactions. A classical example of interacting populations is shown in Figure 4.10.30 for the snowshoe hare and the Canadian lynx, a specialist predator. The lynx-hare... 

Vito Volterra (1860-1940), an Italian mathematician, and Alfred J. Lotka (1880-1949), an American mathematical biologist, formulated at about the same time the so-called Lotka-Volterra model of predator-prey population dynamics. The assumptions of this model are ... 

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