Lotka

Lotka, A. (1925). Elements of Physical Biology. Baltimore Williams Wilkins. 

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. 

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

FIGURE 2.6 Population dynamics predicted by the Lotka-Volterra model for an initial population of 100 rabbits and 10 lynx. 

The Lotka-Volterra reaction described in Section 2.5.4 has three initial conditions—one each for grass, rabbits, and lynx—all of which must be positive. There are three rate constants assuming the supply of grass is not depleted. Use dimensionless variables to reduce the number of independent parameters to four. Pick values for these that lead to a sustained oscillation. Then, vary the parameter governing the grass supply and determine how this affects the period and amplitude of the solution. 

The population growth rate of an organism can be calculated from the Euler-Lotka equation... 

One of the simplest schemes, which describes the autooscillation regime, was considered by Lotka [224]. 

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... 

Figure 3-34. Lotka-Volterra s predator and prey kinetics .

Figure 3-35. The concentration of wolves plotted versus the concentration of sheep in the Lotka-Volterra predator-prey kinetics.

The dynamics of controlled systems is an open problem that has recently attracted the attention of scientific community [13]. In fact, oscillatory behavior in chemical systems is an interesting topic (which has been typically studied in autocatalytic reactions, e.g., the Lotka system see [44] and references therein). Dynamics of controlled systems can be explained in terms of interconnections. Indeed, by analogy with control systems, autocatalytic chemical systems can be described as examples of chemical feedback [44]. 

Lotka s intrinsic rate of growth of the population. At an initial position and time, a neutral mutation occurs and afterwards no further identical mutations occur (infinite allele model). We are interested in the time and space dependence of the local fractions of the individuals, which are the offspring of the individual that carried the initial mutation. The goal of this analysis is the evaluation of the position and time where the mutation originated from measured data representing the current geographical distribution of the mutation. We limit our analysis to one-dimensional systems, for which a detailed theoretical analysis is possible. Eqs. (39) and (40) turn into a simpler form ... 

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]. 

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. 

Chemistry Basic Elements Lotka-volterra Mechanism ... 

Fig. 1.29. The steady state in the Lotka model. Control parameter p = 0.2.

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

The Lotka model is based on the following reaction chain... 

Fig. 2.1. A stable node. Four solutions of the Lotka model, equations (2.1.22)—(2.1.23) with the distinctive parameter Kp/401 = 2 are presented. The starting point of each trajectory is...

The use of (2.1.14) gives the Lotka model a single solution for the stationary point... 

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