3.3 Constrained Growth

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3.3 Constrained Growth
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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
  environments carrying capacity.

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Revised Model

• Far from carrying capacity M, population P
  increases as in unconstrained model.
• As P approaches M, growth is dampened.
• At PM, birthrate deathrate dD/dt, so
  population is unchanging

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Revised Model

• So we can revise the growth model dP/dt

births
deaths

• Or

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The Logistic Equation

• Discrete-time version

• Gives the classic logistic sigmoid (S-shaped)
  curve. Lets visualize this for P0 20,
• M 1000, k 50, in (wait for it) Excel!

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The Logistic Equation

• What if P starts above M?

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The Logistic Equation
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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.

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(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 gt 0
• The solution is unstable if no such interval
  exists.

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Stability Visualization
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Instability Visualization
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Stability Convergent Oscillation
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Instability Divergent Oscillation