1. Growth and decline

2. Exponential growth pop. size at time t+  t = pop. size at time t + growth increment N(t+  t) = N(t ) +  N Hypothesis:  N = r N  t r - rate constant of growth

3. Differential equation for exponential growth

4. 02468101214161820 0 1 2 3 4 5 6 7 8 9 10 Time - t N(t) Exponential growth r=0.1

5. Exponential growth in discrete time N t+1 = N t + r N t N t+1 = (1+r) N t N t = (1+r) t N 0

6. Exponential decline r - mortality rate

7. Time – t N(t) Exponential decline r=0.1

8. Limited growth Factors that affect population dynamics reproduction (growth rate) mortality environmental capacity

9. Monomolecular model for limited growth First order chemical reaction: A  P A – reactant, P – product, R(t) – reactant concentration k – reaction rate - Exponential decay C(t) – product concentration A = R(0)

10. 012345678910 0 5 15 20 25 30 35 40 45 50 Concentrations time Product C(t)=A(1-exp(-kt)) Reactant R(t)=A exp(-kt) Monomolecular growth

11. Logistic growth model Relies on the hypothesis that population growth is limited by environmental capacity K – environmental capacity

12. 02468101214161820 0 50 100 150 time N(t)

13. Logistic growth with time delay Factor that limits growth acts after some time T D No analytical solution

14. 0102030405060708090100 0 200 400 600 800 1000 1200 1400 1600 1800

15. Discrete logistic model

16. Growth of individual organisms

17. Von Bertalanffy’s model Postulates: Gain in weight is proportional to the surface area of the organism Loss in weight is proportional to the weight of the organism Organism maintain the same shape while growing

18. Von Bertalanffy’s model S – surface area W – weight L - length H,C - parameters (monomolecular growth)

19. Richards’ family of models Has all of previous models as special cases

20. Allometric growth Allometry – study of relative sizes of different parts of organisms X, Y Hypothesis:

21. Computations Matlab script files and functions Simulink block diagrams

22. Computations Matlab functions: exp(x) - exponential plot(x,y) - plot ode45 – compute solution to ODE X=A\B - least squares (help slash) fmins - minimize function over arguments