Prey-predator system Lotka—Volterra model

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 

When the birth rate equals the death rate in the prey (host) population, so that j8 = 0, we have 

The trajectories show that the prey population is minimum at T = j3/a. 

To investigate the stability properties of the Lotka-Volterra equation in the vicinity of the equilibrium point (X, Y) = (0, 0), we linearize the equations ofXand Yappearing on the right side of Eqs. (13.63) and (13.64). These functions are already in the form of Taylor series in the vicinity of the origin. Therefore, the linearization requires only that we neglect the quadratic terms in XT, and the Lotka-Volterra equations become 

The first equation shows thatX increases exponentially, and hence the equilibrium point (0, 0) is unstable. A relationship between the X and T can be obtained by the method of separation of variables, which yields 

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

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... 

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. 

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. 

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]. 

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