BIO3011 Lecture 11: Week 10. Equilibrium and stability

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10. Equilibrium and stability
Dynamic equations often have one or more equilibrium states
An equilibrium refers to a situation in which the dynamic system does not change over time
An equilibrium will occur:
o Disete tie odel: N’ = N
o Continuous time model: dN/dt = 0
When either of these are true, the population size does not change
When the conditions are false, the population does change
When the recursion is larger than the variable, there is an increase in variable over time
i.e. When p > 0, Wolbachia increases in the population; when p < 0, it decreases.
Equilibrium is when a system reaches a state of the variable/s in which no further change
will occur
Stable = when a system that is near equilibrium will move closer to equilibrium
Unstable = when a system near equilibrium moves away
Wolbachia:
o At euiliiu p’=p
o p= sf/sh (equilibrium)
o p hat = 0 = 0, p hat = 1 = 1
o go over 10:49
o -1< slope <1 = stable
o If slope is positive but less than one = stable
o If slope is > 1 = unstable
o P is variable
o Wolbachia.rec is function
Plotting recursion allows us to see how many equilibria there are (where the lines overlap)
Why are we plotting between 0 and 1 -> most biologically relevant points -> eg. talking
about frequency alleles in wolbachia where 1 = 100%1
Density dependent model:
o left point is unstable, right top is stable
o when r is large very complex, starts jumping around
o r = growth rate
Natural selection model:
o eusio is alays aoe efeee lie p’>p
o always get increases in frequency
o 2 equilibria, 0 is unstable, 1 is stable (globally stable)
o If AA is ost, aa is est = goes toads 0, p’<p
P<-1 = stable (at 0)
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