Simulating exponential growth

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Simulating exponential growth
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Exponential growth model

• Assumptions
• r is constant (, -, or 0)
• Population is closed
• No genetic variation
• No age/size structure
• No time lags
• When does exponential growth occur?

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Logistic growth model

• First modification We let r vary
• rmax max. difference between birth and death
  rate
• K carrying capacity or when dNt/dt 0

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Developing the logistic

• Make birth and death rate functions of density
• Linear Compensatory functions
• Consider equilibria
• Define rmax
• Define K
• Derive logistic equation

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Population Equilibrium points

• Unstable if disturbed, population tends to move
  away from original equilibrium state
• Stable if disturbed, population tends to return
  to original equilibrium state

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Ball on a surface analogy
Unstable
Globally, asymptotically stable
Stable, but not asymptotically stable
Locally, but not globally stable
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Types of equilibria
locally stable
locally stable
Non-stable
globally stable
non-equilibrium
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(No Transcript)
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Two equilibrium points
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Behavior of the Logistic growth model

• If Nt lt K, then change in Nt ??
• If Nt K, then dNt/dt ??
• If Nt gt K, then dNt/dt ??
• Nt will approach K asymptotically from above and
  below

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Comparing exp logistic growth dN/dt vs N
lo pc gr
hi pc gr
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Comparing exp logistic growth per capita
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Exponential vs Logistic growthN vs t
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Solving the logistic
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Behavior of the logistic
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Simulating logistic growth
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Birth or Death rate can be density-dep
What if both Density-indep?
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Equilibria can change due to changes in
Density-dependence
Remember
Density-independence
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In reality
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Population regulation and density-dependence

• Compensatory factors necessary for equilibrium
• birth or death must be den-dep
• Equilibrium can change due to
• Strength of compensation
• Compensatory and extrapensatory factors
• Importance of factors can vary
• organism
• location

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Logistic growth model

• Assumptions
• All individuals alike
• No time lags
• K rmax constant
• Linear relation between r and N
• Potential Modifications
• Add time lags
• Add curvilinearity
• Add stochasticity
• Incorporate age and size structure

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Modifying the logistic model

• Incorporating time lags
• 1. Reaction time lag
• Delay differential model
• Length of time lag (tau)
• Response time (1/r)

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Modifying the logistic model

• Incorporating time lags
• 2. Gestation time lag
• Tau vs r

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Modifying the logistic model
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Modifying the logistic model

• Incorporating curvilinear relationships
• Complex functions-not easily expressed

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Modifying the logistic model

• The discrete logistic
• Built-in time lag (?Nk) complex dynamics with no
  stochastic elements

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Modifying the logistic model

• The discrete logistic (integrated)
• Time lag ?Nk1 between Nk
• rmax can controls dynamics

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Dynamic behaviors of the logistic
Damped oscillations
4-point limit cycles
chaos
2-point limit cycles
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What is chaos?

• Irregular fluctuations
• NOT random change (i.e. pattern is repeatable)
• Source is
• Density-dependent feedback AND
• Time lag

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Simulating logistic growth

• Incorporating time lags

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Introducing stochasticity
P?10.5 (h) P?30.5 (t)
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Introducing stochasticity

• Environmental (varying r)
• Good and bad years

No20 ri0.05 Var ri0.0001
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Introducing stochasticity

• Demographic
• Reproduction is discrete
• Deterministic BBDBBDBBD
• Stochastic BBBDDBDBBBBD

No20 b0.55 d0.50
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Introducing stochasticity

• Demographic
• Especially important at small population sizes

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Summary

• Fluctuations due to
• Changes in environment, or
• Random birth and death
• Population can go extinct even if rgt0
• Risk of extinction greater at ?

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Introducing stochasticity

• Logistic growth
• Not Chaos (due to ?)
• Let No vary

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Introducing stochasticity

• Let K vary
• Average N always smaller than K (why?)
• Variable environment--smaller average N
• If large r, sensitive to variation in K

hi r tracks variation
low r averages variation
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Simulating logistic growth

• Incorporating stochasticity