Literature

• [KH] S. Komineas, E. Harmandaris, ”Mathematical Modeling” (Kallipos repository)
• [Mu] R.D. Murray, ”Mathematical Biology” (Sec. 10, Sec. 11.1, 11.2)

Population models [Video lecture]

We assume a population $N=N(t)$ at time $t$ of some animal or other living species (e.g., birds or flies).

Malthus model (Bibliography. [Mu] Sec. 1.1, [KH] Sec. 5.2.1)

Assumptions

1. There is an infinite amount of food.
2. There are no predators.

Assume that we have $b$ births per bird per unit time. [We assume $b$ is constant.]

The population in the next time instance, after time $\Delta t$, is

$$ N(t+\Delta t) = N(t) + b N(t) \Delta t \Rightarrow \Delta N = N(t+\Delta t)-N(t) = b N(t) \Delta t. $$

For infinitesimal time $\Delta t \to dt$, we have the differential equation

$$ \frac{dN}{dt} = \lim_{\Delta t \to 0} \frac{\Delta N}{\Delta t} = bN. $$

[This is the population model of Malthus. It is a linear differential equation. Also, it is a dynamical system giving the dynamics in time of $N$.]

The population is given by the solution of the equation,