Understanding Population Growth Models: Exponential & Logistic

2. Births Demography: the study of these processes Population size or Density “state variable” Immigration Emigration General model of population growth: Nt+1 = Nt + Bt – Dt + It – Et X X Deaths

3. Begin: Simple Model • All individuals identical • Define replacement rate as R (sometimes λ) • Example: • If R = 3, what is the population size at t=4? • N4 = N0* R* R* R* R • N4 = N0* R4 • N4 = N0 34 • Nt = N0 * Rt

4. Exponential growth • Discrete Time model: Populations reproduce only at limited times— • Nt= N0Rt • How do we describe the RATE of change with time? • Nt+1 = NtR • Ris the replacement rate (or lambda) • λ= Nt+1 /Nt • Density-independent • (R does not change with pop size) • Resources not limiting “difference equation”

5. Increasing the R(or lambda)

6. Begin: Simple Model • Define replacement rate as R (sometimes λ) • Example: • If R = 3, what is the population size at t=4? • N4 = N0* R* R* R* R • N4 = N0* R4 • N4 = N0 34 • Nt = N0 * Rt (discrete exponential) • 3 = e1.099 • ...thus R = er where r is intrinsic growth rate • Nt = N0ert (continuous exponential)

7. Exponential growth • Continuous time model: Population size changes continuously • Nt = N0ert • ris the intrinsic growth rate (per captia change) • The population growth rate is: • dN/dt = rN “differential equation” • Density-independent (r does not change with pop size) • Resources not limiting

8. General models: Exponential Growth • Describe how idealized populations would grow in infinite environments… • Is the population growing “un-checked” over the short term? • If yes, then density-independent model may be reasonable approx. • Two forms: • Geometric • Exponential Discrete Continuous • Two options: 1) increase to infinity or 2) decrease to zero….just a matter of time (rate)

9. Okay, but we know most populations don’t grow unchecked! As populations grow what happens to demographic rates? • Intuition? Death Rate Birth Rate N

10. Limits on Population Growth • Density Dependent Limits? • Food/prey • Water • Shelter, nest sites, territories • Disease • Mates • Density Independent Limits? • Weather • Includes stochastic events: hurricanes, fires • Climate • But sometimes climate effects become density dependent….example: El nino in the Galapagos Is.

11. Logistic Population Growth • Exponential population growth with a linear decrease in r as a function of N • growth rate diminishes as limit is approached • Carrying capacity (K) = max # individuals that can be supported in the environment • dN/dt = r0N(1- N/K) r0is the equivalent of the intrinsic growth rate rrealized = r0 (1-N/K) rate of growth slows (linearly) to zero as K is reached N 1- 1000

12. Adding non-linear feedback Non-linear effects of density (N) on r Theta logistic model dN/dt = r(1- (N/K)θ) Where θvaries from 0 to infinity (shape parameter) When θ = 1, linear function (same as exponential)

14. How to recognize density dependence • Manipulate density of an organism • Record individual performance across a range of densities (growth, survival, reproduction) • Pearl (1927) as a classic example • Or, observe the success of individuals as a function of the number of adults. • Examples-reproduction: • Fisheries stock-recruit relationships

15. One of the first laboratory ‘tests’ Pearl (1927) • Maintained Drosophila colonies in bottles with fixed amount of yeast

16. Classic (Ricker) stock-recruit model

18. N t dN dt N Ways to look at simple dynamics of populations(density dep. & density indep.) 1. Time series • Number of individuals (N) at each time t 2. Population rate of change • dN/dt = Nt+1-Nt # New added versus pop’n size (N) 3. Per capita rate of change • dN/dt/N = (Nt+1-Nt)/Nt Does pop’n growth rate change with N? *Exponential example dN dt N N

19. Logistic

20. Logistic

21. Logistic New recruits each timestep Number

22. Logistic

23. Logistic Rate of pop’n growth per individual (NOT CONSTANT) Number

24. Logistic Population Growth Population abundance (N) Time Highest population increase at intermediate densities dN/dt Density (N) Declining per capita contribution dN/dt/N Density (N)