Logistic Dynamics

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Logistic Dynamics
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Lecture Goals

• Define and discuss different perspectives on
  intraspecific competition
• Understand terms for discussing model behaviour
• Infer model behaviour from both discrete and
  continuous models
• Become familiar with an alternative discrete
  logistic model

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Intraspecific Competition(The Source of K)

• Competition Ultimately Reduces Fitness
• feeding (energy for the following)
• breeding (fecundity offspring)
• survival (to continue to do the above)
• Resource must be limited
• not usually O2 , temperature
• could be nest holes, sunlight, water
• Reciprocity in interactions
• both reduce the success of each other
• though one may win, either could
• Density dependence
• more intense with more individuals

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Scramble vs. Contest Competition

• little/no competition at low densities
• as density increases, more resource is used
• above density threshold the resource becomes
  limiting
• not enough for all to get their optimal amount

Contest
Scramble

• Resource is divided equally
• Eventually none get enoughas population grows

• a fixed number win
• get equal and sufficient resources
• others lose, get nothing
• skill? Lottery?
• musical chairs analogy

(Nicholson 1954)
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Interference vs. Exploitation
Exploitation
Interference

• individuals compete solely by the removal of
  resource
• reduced success is not easily reversed
• resource must renew
• e.g. over-grazed parkland

• interactions among individuals
• crowding
• agonistic behaviours
• impact of foragers on prey (refuge use)
• reversible with a decline in forager density
• e.g. pigeons on a pile of seeds

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Examples SCRAM CONT INTER EXPL Bison X X Dougla
s Fir X X Seabird Colony X X on Island
What is the relevant resource for each of
these? Why are they classified in the above
manner?
Bison Range grasses Douglas Fir sunlight /
soil Seabird Colony on Island nest space on rocks
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Terms for Model Dynamics
Variables values that change through the
execution of a model Parameters values that
determine models behaviour and remain constant
through its execution
Variables vary, parameters dont.
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Terms for Model Dynamics
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The Roller Coaster Analogy
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Equilibria in Difference Equations
Nt1 Nt Ne
???
Ne l Ne
Geometric
Ne 0 when l ? 1 or Ne ? ? when l 1 (all real
numbers)
What about the discrete logistic can we prove
what we know?
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Nt1 Nt R 1 (R - 1)/K
Nt
Ne Ne R 1 (R - 1)/K
Ne
Ne 0, but what else?
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R 1 (R - 1)/K Ne
R 1 (R - 1)/K Ne
K Ne
Q.E.D.
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Discrete Logistic An Alternative formulation
Nt1 l Nt Nt er
So we use the density dependent term of the
logistic to modify the population growth rate
Nt1 Nt er (1-Nt/K)
This form allows a more direct comparison to the
continuous logistic model due to its similar form.
Equilibria?
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Ne Ne er (1-Ne/K)
Obviously, Ne 0 works again and also
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Stability?
To investigate stability

• Select an equilibrium value
• Calculate the next value following a perturbation
  above and below the value
• Determine if next value moves toward (stable) or
  away (unstable) from the equilibrium

For example, K in the previous model
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Nt1 Nt er (1-Nt/K), let Nt (K1)
Nt1 (K1) er (1- (K1) /K) focus on l

As (K1)/K gt 1 ? r (1-(K1)/K) lt 0
r (1-(K1)/K) lt 0 ? e r (1-(K1)/K) lt 1
And thus Nt1 lt K1
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• Similarly, when N t K 1, N t1 gt K - 1
• This implies a tendency to remain at K
• difference equations can be more complex
• particularly when the maximum population growth
  rate is high
• to be seen lectures assignment

What about stability around Nt 0?
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Equilibria and Stability in Differential Equations
The arguments are easier to follow with rates!
Equilibrium no change ? dN/dt 0
For all possible N dN/dt r N (1 N/K)
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Stability?
dN/dt

N
0
K
-
0
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Equilibria and Stability in a Novel Model
dN/dt something very ugly
dN/dt
N
0
Stability? How do you draw this?
0