Predator-prey model

Diffusive instability can appear in simple predator-prey models. Bartumeus et al. (2001) used the linear stability analysis and demonstrated that a simple reaction-diffusion predator-prey model with a ratio-dependent functional response for the predator can lead to Turing structures due to diffusion-driven instabilities. 

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. 

LMtka-Volterra predator-prey model. Here R t) is the number of rabbits, F t) is the number of foxes, and a,b,c,d>0 are parameters. 

F. Bartomeus, D. Alonso, and J. Catalan. Self-organized spatial structures in a ratio-dependent predator-prey model. Physica A, 295 53, 2001. 

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). 

The phase portraits in Figs. 15.6 and 15.7 are reminiscent to those of other excitable systems. This is even more so if the characteristic time scales of the prey are fast compared to that of the predators (as it is typically the case, since prey species usually have smaller biomass than their predators). The time scale separation can be made explicit in the model with the help of a new dimensionless parameter e. This leads to the following excitatory predator-prey model... 

The requirements to describe population outbreaks with a predator-prey model (15.5) are the following (i) sigmoidal (type-III) functional response ... 

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. 

Myerscough, M.R., Darwen, M.J. Hogarth, W.L. 1996 Stability, persistence and structural stability in a classical predator-prey model. Ecol. Model. 89, 31-42. 

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. 

Why a reaction oscillates can be understood by looking at a simplified predator-prey model. If the population of rabbits is taken as the concentration of one reactant, and the population of wolves taken for the other, then if the rabbits food supply is held constant, the rabbit... 

Gilipin, M. (1979). Spiral chaos in a predator-prey model, Amer. Naturalist, 113, 306-8. 

Peschel, M. Mende, V. (1985). The predator-prey model Do we live in a Volterra worldl Akademie Verlag. 

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. 

Sometimes chemical problems can be answered using the knowledge from other sciences that are not related to chemistry at first sight. For example, some information about a complex reactions flow can be gained from the mathematical models of the interspecific competition. A classical example is the predator-prey model, which describes the population trends for predators and prey in living conditions... 

Even though the predator-prey model is rather idealized, many kinetic models for real chemical systems are based on it. For example, D.A. Frank-Kamenetsky used the Lotka-Volterra model to explain the processes of higher hydrocarbon oxidation. 

Fig. 3.15 Predator-prey model analysis using Maple...

This mechanism can show oscillatory behavior if A is continually replenished so that [A] remains constant This mechanism has been used in ecology as a simple predator-prey model, in which A represents the food supply (grass) for prey animals (hares), represented by X. Predators (wolves) are represented by Y, and dead wolves are represented by P. The consumption of grass (A) by the hares (X) allows them to reproduce as in step 1, and the consumption of hares by wolves (Y) allows the wolves to reproduce as in step 2. Step 3 corresponds to the death of wolves by natural causes. Since no... 

M. Gatto, S. Rinaldi Stability analysis of predator-prey models via the Liapunov method. Bull. Math. Biol. 39, 339 (1977)... 

W. C. Chewning Migratory effects in predator-prey models. Math. Biosci. 23, 253 (1975)... 

Another interesting application is the use of evolving network models for a simulation approach to hard game-theoretical problems such as predator-prey models in continuous space-time. They cannot simply be reduced to discrete tournaments. One can find solutions to such problems by simulating the players as evolving neural nets [51]. In this model, the universal emergence of the main prey behavioral patterns observed in nature was verified in artificial evolution. The three stages are ... 

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