Malthusian population growth

The term kj X (k > 0) in (2.44) describes the reproduction of the prey species due to a constant supply from some external food resource. Without a predator species being present, the prey X would exponentially increase in population (Malthusian population growth). The k -term (k > 0) represents the loss rate of the prey species due to "collisions" with the predators. The ansatz for this loss rate proportional to the product of population densities XY may be understood by comparing it with a collision process in reaction kinetics. Accordingly the k -term (k > 0) represents the gain rate of the predator species, whereas the k2"term (k2 > 0) describes that the predator species would exponentially die out in the absence of preys. 

Malthusian population growth. The rate of growth dp/dt of a biological population is often proportional to the population pit) itself, because the population size determines the number of parents that can beget offspring dp/dt = oip. Show that growth is exponential if cx > 0. 

Autocatalysis, with n = 1, characterizes biological population growth, for example, since the number of offspring born is proportional to the number of individuals in the population. It leads to the Malthusian population explosion. In chemical systems, where autocatalysis is less common, it can also result in explosion, since the solution to Eq. (2.1) is an exponentially growing concentration. Of relevance to polymer systems is the fact that any exothermic reaction is inherently autocatalytic, since an increase in the product concentration corresponds to production of heat, which leads to an increase in the rate constant of the reaction via the Arrhenius factor. If the reaction in question is lengthening a polymer chain by addition of the monomer, the rate should increase as the chain grows if the heat produced is not rapidly removed from the system. 

As an example we may present logistic model of growth (8), where F = N - current value of population number r - Malthusian parameter Fa, =N00 - capacity of medium (limited value of N). 

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