Volterra Predator-Prey Systems

We can model this behaviour with a set of three reactions and their differential equations, (a) In the first reaction the sheep are breeding. Note, that there is a constant supply of grass and this reaction could go on forever. As it is written, this reaction violates the law of conservation of mass, it is only an empirical rate law. In a second reaction (b), wolves eat sheep and breed themselves. The third reaction (c) completes the system, wolves have to die a natural death. 

Surprisingly, the dynamic of such a population is completely cyclic. All properties of the cycle depend on the initial populations and the rate constants . This behaviour is best seen in a plot of the wolf vs. the sheep concentration. For any set of initial concentrations and rate constants , this cyclic behaviour is maintained. 

Note the imperfect coincidence of the line. This effect is due to small numerical errors increasing the accuracy of the solver reduces these differences. 

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Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

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... 

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... 

Example 13.8 Prey-predator system Lotka-Volterra model The Lotka-Volterra predator and prey model provides one of the earliest analyses of population dynamics. In the model s original form, neither equilibrium point is stable the populations of predator and prey seem to cycle endlessly without settling down quickly. The Lotka-Volterra equations are... 

The classical theory of predator-prey interaction as formulated by Volterra involves two equations which express the growth rate of the prey and the predator (57). Within the context of phytoplankton and zooplankton population, the prey is the phytoplankton and the predator the zooplankton. In the notation of the previous sections, for a one-volume system, the Lotka-Volterra equations are ... 

Fig. 15.2. Predator-prey cycles in Lotka-Volterra system (15.1). Left neutrally stable center in the phase plane. Oscillations are seen as closed loops of orbits around the center. Different initial values lead to different trajectories. Right typical time series of predator N t) (solid line) and of prey R(t) (dotted line).

Autocatalysis is involved in the first two steps of this process. It appears that oscillating chemical reactions have mechanisms that are different from the Lotka-Volterra mechanism. This type of mechanism does occur in certain types of complex system such as predator-prey relationships in biology. It was in the biological context that the mechanism was investigated by the Italian mathematician Vito Volterra (1860-1940). 

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. 

Let us now consider in detail the classical predator-prey problem, that is, the interaction between two wild-life species, the prey, which is a herbivore, and the predator, a carnivore. These two animals coinhabit a region where the prey have an abundant supply of natural vegetation for food, and the predators depend on the prey for their entire supply of food. This is a simplification of the real ecological system where more than two species coexist, and where predators usually feed on a variety of prey. The Lotka-Volterra equations have also been formulated for such... 

From the 1970s onwards, Cesare Marchetti and other system analysts have studied thousands of artifacts, and have discovered that their behaviour is described by the same equations that Lotka and Volterra found for the behaviour of predators and prey. The growth pattern of cars, for example, is a logistic curve. Cars spread in a market exactly as bacteria in a broth or rabbits in a prairie. Cultural novelties diffuse into a society as mutant genes in a population, and markets behave as their ecological niches. But why ... 

Example 13.9 Prey—predator system—Lotka—Volterra model... 

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 ... 

The system rotates irreversibly in a direction determined by the sign of /. An example of such a system is the well-known Lotka-Volterra prey-predator interaction given as an exercise (exc. 18.9). We can also apply this inequality to derive a sufficient condition for the stability of a steady state. If all fluctuations fipP > 0 then the steady state is stable. But here it is more expedient to use the Lyapunov theory of stability to which we turn now. 

Apparently, a number of factors affecting the development of the populations of the predators and their preys in reality id much larger, than is considered by Lotka-Volterra equations. Assume that the hunters invaded this isolated system. Part of them hunts the prey and constantly kills some quantity of the preys with speed ri. Others kill the predators with speed r2- Investigate, how the interference of the hunters influences the dynamics of the development of both populations. 

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