1. 3.3 Constrained Growth

2. 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.

3. 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:

4. Revised Model So we can revise the growth model dP/dt: Or births
deaths Or

5. The Logistic Equation Discrete-time version:
Gives the classic logistic sigmoid (S-shaped) curve. Let’s visualize this for P0 = 20,
M = 1000, k = 50%, in (wait for it…) Excel!

6. The Logistic Equation
What if P starts above M?

7. The Logistic Equation

8. Equilibrium and Stability
Regardless of P0, 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.

9. (Un)stable: Formal Definitions
Suppose that q is an equilibrium solution for a differential equation dP/dt or a difference equation DP. 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
P(t) is finite for all t > 0
The solution is unstable if no such interval exists.

10. Stability: Visualizationq b

11. Instability: Visualization

12. Stability: Convergent Oscillation

13. Instability: Divergent Oscillation