Growth and decline Exponential growth pop size at

Growth and decline

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

Differential equation for exponential growth

Exponential growth r=0. 1 10 9 8 7 N(t) 6 5 4 3 2 1 0 0 2 4 6 8 10 12 Time - t 14 16 18 20

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

Exponential decline r - mortality rate

Exponential decline r=0. 1 N(t) Time – t

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

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)

Monomolecular growth 50 45 40 Product C(t)=A(1 -exp(-kt)) Concentrations 35 30 25 20 15 Reactant R(t)=A exp(-kt) 10 5 0 0 1 2 3 4 5 time 6 7 8 9 10

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

150 100 N(t) 50 time 0 0 2 4 6 8 10 12 14 16 18 20

Logistic growth with time delay Factor that limits growth acts after some time TD No analytical solution

1800 1600 1400 1200 1000 800 600 400 200 0 0 10 20 30 40 50 60 70 80 90 100

Discrete logistic model

Growth of individual organisms

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

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

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

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

Computations Matlab script files and functions Simulink block diagrams

Computations Matlab functions: exp(x) - exponential plot(x, y) - plot ode 45 – compute solution to ODE X=AB - least squares (help slash) fmins - minimize function over arguments