2.3 Constrained Growth

Carrying Capacity Exponential birth rate eventually meets environmental constraints (competitors, predators, starvation, etc.) Maximum population size that a given environment can support indefinitely is called the environment’s carrying capacity.

Revised Model Far from carrying capacity M, population P increases as in unconstrained model. As P approaches M, growth is dampened. At P=M, birthrate = deathrate dD/dt, so population is unchanging. First, define dD/dt:

Now we can revise the growth model dP/dt: Revised Model births deaths Or:

The Logistic Equation Discrete-time version: Gives the classic logistic sigmoid (S-shaped) curve. Let’s visualize this for P 0 = 20, M = 1000, k = 50%, in (wait for it…) Excel!

The Logistic Equation What if P starts above M?

The Logistic Equation

Equilibrium and Stability Regardless of P 0, P ends up at M: M is an equilibrium size for P. An equilibrium solution for a differential equation (difference equation) is a solution where the derivative (change) is always zero. We also say that the solution P = M is stable. A solution with P far from M is said to be unstable.

(Un)stable: Formal Definitions Suppose that q is an equilibrium solution for a differential equation dP/dt or a difference equation  P. The solution q is stable if there is an interval (a, b) containing q, such that if the initial population P(0) is in that interval, then 1.P(t) is finite for all t > 0 2. The solution is unstable if no such interval exists.

Stability: Visualization q a b

Instability: Visualization

Stability: Convergent Oscillation

Instability: Divergent Oscillation