## Dynamics of One Population

Perhaps the first example from mathematical ecology was considered by Leonardo di Pisa (known as Fibonacci). In 1202, he published a book entitled Liber Abacci in which he stated and solved the following problem. Suppose a man puts into a space surrounded by walls one pair of rabbits. Each month a pair of rabits produces a new pair, while a newborn pair needs two months to produce a pair of rabbits. How many pairs will be there after one year? (Answer: 377 pairs.) Fibonacci assumed that at the start, the pair of rabbits put in the enclosure is a mature pair. His series goes as follows: 1 (start), 2 (during the first month), 3 (during the second month), 5 (during the third month). Number of pairs of rabbits each following month is a sum of numbers of previous two months, so his series ends with 377 pairs after the 12th month. Kepler (1611) showed that the problem of Fibonacci can be represented by the following second-order difference equation: N(t + 2) = N(t + 1) + N(t) with N(0) = 1 and N(1) = 1.

Another early proposal for population growth came from Malthus in his monograph An Essay on the Principle of Population, 1798. Today, we call his proposal the first law of population growth. It describes a geometric law of population growth if we adopt the discrete representation by the first-order difference equation,

The parameter rd is the so-called biotic potential which stands for the difference between the numbers of births and deaths during 1 time interval. If we adopt the continuous formulation, the Malthus law may be expressed by the first-order ordinary differential equation:

and it describes an exponential growth of the population. In order that the geometric growth coincides with the exponential growth, the condition rc = ln(1 + rd) must be satisfied. The parameter rc is smaller than rd (except in the trivial case: rc = 0 = rd) because rc is an instantaneous biotic potential. Both geometric and exponential growth are expected to hold in an environment where the population number is large enough but far below the environmental capacity, that is, the population that the environment can support. The same law, but with negative rc value, was formulated by Rutherford in 1900 to describe the radioactive decay of thorium. The law was also proposed by Chick in 1908 to describe disinfection kinetics.

The next conceptual step further came in 1846 when Verhulst formulated what is called today the second law of population growth. The law can be represented by the following nonlinear ordinary differential equation:

where K is a constant known as the carrying capacity of the environment. As stated above, K specifies the number of organisms that the sources of food in the environment can support. When the starting population is much smaller than K, the solution of the eqn  looks like a sigmoidal curve. The curve has been termed logistic curve and the growth of the population a logistic growth.

The equivalent difference equation is N(t + 1) = N(t) + rd N(t) (1 - N(t)/K). As the value of the biotic potential increases beyond 2, the number of stable equilibrium points grows to infinity, thus producing what has been termed a chaotic dynamics. For values equal to 3 and higher, the extinction of the population is imminent. This is the subject where systems ecology and mathematical theory of chaos are linked. Once chaotic dynamics has been investigated, it leads to the control theory of chaos with important applications for management of epidemics, physical phenomena, and medicine. May in 1976 was the first in ecology to draw attention to this problem and published a number of important scientific advances on the subject.

Dynamics of one population has initially been analyzed in a peaceful environment that is, when K is constant. However, the source of food in the environment may be essentially periodic, for example, expressing the effect of seasonality. In that case, at least the parameter K must be replaced by a periodic function, for example, K(t) = Ko + K1 sin !t. Here Ko is the average value of K(t), K1 is the amplitude of periodic variation, and ! is a frequency equal to 2'k/T (where T is the period). In case that rT<< 1, the dynamics is qualitatively similar to the logistic growth except that the population does not tend to Ko but to -y/(Ko2 — K12). If rT>> 1, the dynamics has an initial transient part which may roughly resemble the logistic growth but ends up in the periodic fluctuation: N(t) « Ko + Ki sin !t. In nature, rT is in between the two extremes, and hence we expect the population will vary periodically, but the average population number will be smaller than in the case of constant environment. It may also be the case that the biotic potential r(t) is periodic. In this case, the dynamics will be more complicated but the average population number will again be smaller than Ko.

Population dynamics has also been investigated in a purely random environment where either r = ro + 7X0, cr2) or K = Ko + 7K(0, cK2) or both. Here ro and Ko are average values of the biotic potential and carrying capacity, respectively, while the two 7 functions are random functions or noises with average value 0 and corresponding variances c2. If the population starts with a value which is much smaller than Ko, the dynamics will first have a transient part which may roughly resemble a sigmoidal growth. It will then tend to a noisy dynamics. If either cr2 or cK2 or both are high, the population fluctuation will be large. In this second part ofthe dynamics, the average value of the population will be smaller than Ko and it will decrease as any of cr2 or cK2 increases. Finally, as cr2 or aK2 increases further, population will tend to extinction.

When organisms reproduce in phase and there is a significant delay between conception and birth, eqn  or any equation where the right-hand side is a function of N only, cannot reproduce population dynamics. Then it is necessary to introduce a delay so that the right-hand side is also a function of N(t — r), where r is the time between conception and birth. Depending on the values of r and r, the dynamics may resemble logistic growth, tending with damped oscillations to a constant value, or a periodic variation. Finally, if r and r are large enough, population number will periodically get very close to zero, that is, the population will periodically face a real threat of extinction.

In some species, younger organisms feed on an entirely different source than older organisms. To analyze dynamics of such populations it is necessary to treat subpopulations according to age or stage groups. Models become more complicated, analyses more difficult, and results more specific to a particular species and area. However, modeled population dynamics is closer to reality and enables finer understanding of consequences following environmental perturbations. As a result, it enables one to find more subtle ways of population control.

Further, there exist subpopulations of the same species that are spatially separated but experience exchange of organisms to some extent. Such a system of subpopulations is called a metapopulation. Investigation of metapopulation models led to conclusion that following unexpected and large perturbations in the environment, a metapopulation in nature is more likely to persist than a single population. 