Ecology: Biology of Interaction. IV-04. Exponential and Logistic Population Growth
In the logistic model, the variable K is introduced: environmental carrying capacity, the equilibrium population size at which all available resources are consumed. Growth in the logistic model is described by dN/dt = r × N × (K−N)/K. For historical reasons, r can be called the Malthusian parameter.
IV-4. Exponential and Logistic Population Growth. There is only as much real science in every branch of natural history as there is mathematics in it. Immanuel Kant, 1786. " In 1536, the Spanish adelantado Pedro de Mendoza, founding the city of Buenos Aires, brought 20 cows and 72 horses to the Argentine Pampas. Three years later, the settlement was burned to the ground by the Indians, and the Spaniards abandoned it. The horses and cows were left to survive on their own. They multiplied in the Pampas, and by 1700, the population of cows and the population of horses each reached a million heads. Spanish sailors of the 16th and 17th centuries systematically brought goats to oceanic islands to provide for themselves in case of shipwreck. One such traveler, Juan Fernández, brought a pair of goats to the Pacific islands near the coast of Chile - islands that were later named after him. In 1704, when Alexander Selkirk (who served as the prototype for Daniel Defoe's Robinson Crusoe) was left on these islands by his ship's captain, the herd of goats that originated from this pair exceeded 10,000, and the herd still exists today" (O. Solbrig, D. Solbrig, 1982). The population growth is proportional to the number of individuals in it, i.e., ΔN ~ N, where N is the population size, and ΔN is its change over a certain period. If this period is infinitesimally small, it can be written as dN / dt = r × N, where dN / dt is the change in population size (growth), and r is the reproductive potential, a variable that characterizes the ability of a population to increase its size. The given equation is called the exponential population growth model (Fig. IV-4.1). Fig. IV-4.1. Exponential growth. The value r is sometimes called the Malthusian parameter. The English clergyman Thomas Malthus was the first to notice that the population grows in a geometric progression. It was acquaintance with his work that prompted both Charles Darwin and Alfred Russel Wallace to hypothesize that the offspring of any organisms must be "thinned out" by natural selection. As is easy to understand, with the growth of time, the population size increases faster and faster, and quite soon tends towards infinity. Naturally, no habitat can withstand a population of infinite size. However, there are a number of population growth processes that, within a certain time frame, can be described using the exponential model. This applies to cases of unlimited growth when a population colonizes an environment with an excess of free resources: cows and horses colonize the pampas, flour beetles colonize a grain elevator, yeast colonizes a bottle of grape juice, etc. Naturally, exponential population growth cannot continue forever. Sooner or later, the resource will be depleted, and population growth will slow down. What will this braking be like? Practical ecology knows a wide variety of options: both a sharp increase in numbers followed by the extinction of the population that has depleted its resources, and a gradual slowing of growth as it approaches a certain level. Slow braking is the easiest to describe. The simplest model describing such dynamics is called the logistic model and was proposed (to describe human population growth) by the French mathematician P. Verhulst as early as 1845. In 1925, a similar pattern was rediscovered by the American ecologist R. Pearl, who suggested that it has a general character. In the logistic model, the variable K is introduced - the carrying capacity of the environment, the equilibrium population size at which it consumes all available resources. Growth in the logistic model is described by the equation dN / dt = r × N × (K-N) / K (Fig. IV-4.2). For historical reasons, the value r in the logistic model can be called the Malthusian parameter (i.e., Malthus's parameter), and K - Verhulst's parameter. Fig. IV-4.2. Logistic growth. As long as N is small, the population growth is mainly influenced by the factor r × N, and population growth accelerates. When N becomes quite high, the population size is mainly influenced by the factor (K-N) / K, and population growth begins to slow down. When N = K, (K-N) / K = 0, and population growth stops. Despite its simplicity, the logistic equation satisfactorily describes many cases that can be observed in nature and is still successfully used in mathematical ecology.