Population Review

Exponential growth N t+1 = N t + B – D + I – E ΔN = B – D + I – E For a closed population ΔN = B – D

dN/dt = B – D B = bN ; D = dN (b and d are instanteous birth and death rates) dN/dT = (b-d)N dN/dt = rN1.1 N t = N o e rt 1.2

Influence of r on population growth

Doubling time N t = 2 N o 2N o = N o e r(td) (td = doubling time) 2 = e r(td) ln(2) = r(td) td = ln(2) / r1.3

Assumptions No I or E Constant b and d (no variance) No genetic structure (all are equal) No age or size structure (all are equal) Continuous growth with no time lags

Discrete growth N t+1 = N t + r d N t (r d = discrete growth factor) N t+1 = N t (1+r d ) N t+1 = λ N t N 2 = λ N 1 = λ (λ N o ) = λ 2 N o N t = λ t N o 1.4

r vs λ e r = λ if one lets the time step approach 0 r = ln(λ) r > 0 ↔ λ > 1 r = 0 ↔ λ = 1 r < 0 ↔ 0 < λ < 1

Environmental stochasticity N t = N o e rt ; where N t and r are means σ r 2 > 2r leads to extinction

Demographic stochasticity P(birth) = b / (b+d) P(death) = d / (b+d) Nt = N o e rt (where N and r are averages) P(extinction) = (d/b)^N o

Elementary Postulates Every living organism has arisen from at least one parent of the same kind. In a finite space there is an upper limit to the number of finite beings that can occupy or utilize that space.

Think about a complex model approximated by many terms in a potentially infinite series. Then consider how many of these terms are needed for the simplest acceptable model. dN/dt = a + bN + cN 2 + dN From parenthood postulate, N = 0 ==> dN/dt = 0, therefore a = 0. Simplest model ===> dN/dt = bN, (or rN, where r is the intrinsic rate of increase.)

Logistic Growth There has to be a limit. Postulate 2. Therefore add a second parameter to equation. dN/dt = rN + cN 2 define c = -r/K dN/dt = rN ((K-N)/K)

Logistic growth dN/dT = rN (1-N/K) or rN / ((K-N) / K) Nt = K/ (1+((K-N o )/N o )e -rt )

Data ??

Further Refinements of the theory Third term to equation? More realism? Symmetry? No reason why the curve has to be a symmetric curve with maximal growth at N = K/2.

What if the population is too small? Is r still high under these conditions? Need to find each other to mate Need to keep up genetic diversity Need for various social systems to work

Examples of small population problems Whales, Heath hens, Bachmann's warbler dN/dt = rN[(K-N)/K][(N-m)/N]

Instantaneous response is not realistic Need to introduce time lags into the system dN/dt = rN t [(K-N t -T )/K]

Three time lag types Monotonic increase of decrease: 0 < rT < e -1 Oscillations damped: e -1 < rT <  /2 Limit cycle: rT >  /2

Discrete growth with lags

May, Science 1. N t+1 = N t exp[r(1-N t /K)] 2. N t+1 = N t [1+r(1-N t /K)]

(1) N t+1 = N t exp[r(1-N t /K)] (2) N t+1 = N t [1+r(1-N t /K)] Logistic growth with difference equations, showing behavior ranging from single stable point to chaos

Added Assumptions Constant carrying capacity Linear density dependence