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 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

7. Logistic Population Models

8. Logistic Population Models • Here is an example of exponential growth

9. 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), given that no resource can sustain unregulated growth in perpetuity

10. 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

11. 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?

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

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

15. Density-dependent factors impacting population dynamics of a planthopper

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 as a population approaches the carrying capacity dN/dt = rN [1-(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 • Now consider the population growth of a species • At some point competition for resources will strengthen, even in the absence of other species dN/dt = rN [1-(N/K)] Intraspecific competition

19. Logistic Population Models Logistic population models can be used to examine the potential impact of interspecificand intraspecificcompetition, as well as predator-prey relationships and/or population management (harvesting) A competitor (or predator) should lower the population numbers of the target species, but by how much?

20. Logistic Population Models • So the equation for population growth is: • Another species (a competitor) has its own population dynamics… • And has the ability to suppress the population of sp1 dN/dt = r1N1 [1-(N1/K1)] dN/dt = r2N2 [1-(N2/K1)] dN/dt = r1N1 [1-(N1/K1) – α1,2 (N2/K1)]

21. Logistic Population Models In this scenario, α represent a ‘competition coefficient’ or a measure of the intensity of competition You can see the growth of species 1 is proportionately depressed by species 2 Equilibrium: dN/dt = r1N1 [1-(N1/K1) – α1,2 (N2/K1)] N1 = K2 * α N2

22. Logistic Population Models Conversely, we can also monitor the population growth of sp2 As well as the impact of sp1 on sp2 dN2/dt = r2N2 [1-(N2/K2)] dN2/dt = r2N2 [1-(N2/K2) – α2,1 (N1/K2)]

23. Logistic Population Models Some important model assumptions: 1) resources are in limited supply (if not, little or no competition, thus no effect) 2) competition coefficients (α and β) and carrying capacities are constants (otherwise too difficult to predict) 3) density dependence is linear (adding individuals yields a strict linear impact; equilibrium of non-linear systems complex)

24. Logistic Population Models It is extremely difficult to get an accurate an accurate estimate of the competition coefficient (α). Why? Remember, most competition is asymmetrical

25. Logistic Population Models One could expand upon this equation and include as many species for which as one could get a reasonable estimate of the actual intensity of competition These same equations could be modified to further capture the effects of other biotic interactions

26. Logistic Population Models There are many other relationships that can be modeled

27. Logistic Population Models