Population Models

1. Population Models • 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

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

3. Population Models 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 and dt = Dt/Nt Nt+1 = Nt + btNt - dtNt Nt+1 = Nt + bNt - dNt

4. Population Models • 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)

5. Population Models • 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

6. Population Models • 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

7. Population Modelsexponential growth (continuous) • 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

8. Population Modelsexponential growth (continuous) • 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

9. Logistic Population Models

10. Logistic Population Models

11. Logistic Population Models • 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

12. Logistic Population Models • 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

13. Logistic Population Models • 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

14. Logistic Population Models • We will explore the behavior of populations as numbers change • There is an equilibrium population size Neq = b-d d’-b’

15. Logistic Population Models • However, is it realistic to think populations will grow exponentially continuously?

16. Logistic Population Models • 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)}

17. Logistic Population Models • To derive the equation for population size requires us to use calculus Nt = K/ 1+ [(K-N0) / N0]e-rt

18. Logistic Population Models