2.3 Constrained Growth. Carrying Capacity Exponential birth rate eventually meets environmental constraints (competitors, predators, starvation, etc.)

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:

PM

Pr=

dt

dD

• Now we can revise the growth model dP/dt:

PM

Pr(rP)=

dt

dP

Revised Model

PM

Pr=

dt

dD

births deaths

• Or: PM

Pr=

dt

dP

1

The Logistic Equation

• Discrete-time version:

PM

Pr=

dt

dP

1

tr=kΔt),P(tM

Δt)P(tk=ΔP

where1

• Gives the classic logistic sigmoid (S-shaped) curve. Let’s visualize this for P0 = 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 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.

(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.

limt →∞

P ( t )=q

Stability: Visualization

q

a

b

Instability: Visualization

Stability: Convergent Oscillation

Instability: Divergent Oscillation