2.3 constrained growth. carrying capacity exponential birth rate eventually meets environmental...
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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