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Population Models

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- What is a population?
- Populations are dynamic
- What factors directly impact dynamics
- Birth, death, immigration and emigration

- in models we frequently simplify things in order to gain a better understanding of how the rest will work
- E.g. a closed vs. open population

- Start with treating time as a ‘discrete’ (geometric population growth) unit rather than continuous (exponential growth)
- Is this realistic? Why or why not?

Nt = Bt –Dt + It –Et

Nt+1 = Nt + Bt -Dt

- Model development
- Consider using per capita rates (individuals)
- Rewrite the equation in terms of per capita rates:
- With constant rates

bt = Bt/Nt anddt = Dt/Nt

Nt+1 = Nt + btNt - dtNt

Nt+1 = Nt + bNt - dNt

- Model is somewhat realistic, but still useful
- 1) provides a good starting point for more complex models (changes rates)
- 2) it is a good heuristic – provides insight and learning despite its lack of realism
- 3) many populations do grown as predicted by such a simple model (for a limited period of time)

- Because this model does NOT change with population size, it is called density-independent
- Furthermore, (b-d) is extremely important
- λ is the finite rate of increase

Nt+1 = Nt + (b – d)Nt

Nt+1 = Nt + RNt

Nt+1 = (1+R)Nt

Nt+1 = λNt

- Doubling time
- Consider R=0.1; Λ=1+R (1.1)

Nt+1 = λNt

Nt double = 2N0

2N0 = λt double N0

Divide both sides by N0 : 2 = λt double

Take the logarithm of both sides: ln2 = tdouble lnλ

Divide both sides by lnλ: ln2 / lnλ = tdouble

7.27 years

- Instantaneous rate of change
- Calculate the per capita rate of pop growth
- Calculate the size of the pop at any time

dN / dt = rN

(dN / dt) / N = r

Nt = N0ert

- Doubling Time

Nt double = N0ert double

Substitute 2N0 2N0 = N0ert double

Divide by N0 2 = ert double

Take natural log ln 2 = rtdouble

Finally divide by r tdouble = ln2 / r

- Similarly this population model will explicitly model birth and death rates
- Will also add in the concept of a carrying capacity (K), and one that is a continuous-time version

- Remember, the geometric model looked like this:
- We can add two new terms to the model to represent changes in per capita rates of birth and death, where b’ and d’ = the amount by which the per capita birth or death rate changes in response to the addition of one individual of the pop(n)

Nt+1 = Nt + bNt - dNt

Nt+1 = Nt + (b+b’Nt)Nt – (d+d’Nt)Nt

- All four parameters (b, b’, d, d’) are assumed to remain constant through time (hence no bt)
- How and why should b and d vary with density?
- Logistic population models can be used to examine the potential impact of interspecificand intraspecificcompetition, as well as predator-prey relationships and harvesting populations

- We will explore the behavior of populations as numbers change
- There is an equilibrium population size

Neq = b-d

d’-b’

- However, is it realistic to think populations will grow exponentially continuously?

- This equilibrium defined is so important, it is called the ‘carrying capacity’
- This model gives us rate of change of population size

dN = rN {(K-N) /K)}

- To derive the equation for population size requires us to use calculus

Nt = K/ 1+ [(K-N0) / N0]e-rt