– Mathematical Model on Human Population Dynamics using Delay Differential Equation –
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Simple population growth models involving birth rate, death rate, migration, and carrying capacity of the environment were considered. Furthermore, the particular case where there is discrete delay according to the sex involved in the population growth were treated.
The equilibrium and stability analysis of each of the cases were considered also. The stability analysis shows that the discrete delays in the population growth lead to instability in the growth.
One of the most generally accepted ideas of population in ecology is that time delays are potent sources of instability in population growth system. If true, this statement has important consequences for our understanding of population dynamics, since time delays are ubiquitous in ecological systems.
All species exhibit a delay due to maturation time. Whenever specie has a recognizable breeding season there can be a lag between an environmental change and the reproductive response of the specie.
They also exhibit a delay due to gestation period and regeneration period. The destabilizing effect of time delays is often expressed by the rule that an otherwise stable equilibrium will generally become unstable if a time delay exceeds the dominant time scale of a system (May 1973a, b; Mayriard Smith 1974).
A dynamic system has two basic time scales namely: the return time, which reflects the rapidity with which the system returns to equilibrium following a small perturbation, and the natural period, which is the period of oscillation exhibited by a perturbed system.
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