1. What is a Population A collection of potentially interacting organisms of one species within a defined geographic area.

2. Figure 14.1

3. Estimates of Population Size Recall the Lincoln index from recitation: M/N = m/n M is the total number marked; m is the number marked in the sample; n is the sample size.

4. Elementary Postulates 1. Every living organism has arisen from at least one parent of the same kind. 2. In a finite space there is an upper limit to the number of finite beings that can occupy or utilize that space.

5. Populations grow by multiplication. A population increases in proportion to its size, in a manner analogous to a savings account earning interest on principal: at a 10% annual rate of increase: a population of 100 adds 10 individuals in 1 year a population of 1000 adds 100 individuals in 1 year allowed to grow unchecked, a population growing at a constant rate would rapidly climb toward infinity

6. Geometric population growth Usually think of animals increasing by distinct generations. N(1) = N(0) + B – D + I – E N(1) = N(0)R N(2) = N(0)RR N(t) = N(0)R t

7. Critical parameter is = net replacement rate (R o ). N(t+1) = N(t) where: N(t + 1) = number of individuals after 1 time unit N(t) = initial population size = ratio of population at any time to that 1 time unit earlier, such that λ = N(t + 1)/N(t)

8. To calculate the growth of a population over many time intervals, we multiply the original population size by the geometric growth rate for the appropriate number of intervals t: N(t) = N(0) t For a population growing at a geometric rate of 50% per year ( = 1.50), an initial population of N(0) = 100 would grow to N(10) = N(0) 10 = 5,767 in 10 years.

9. Given N o = 5000; N 1 = 6000; What is N 2 ? N 2 = (6/5) 2 x 5000 = 7200

10. Exponential Growth Generations overlap, usually not discrete generations. For convenience, most of our models are continuous.

11. Think about a complex model approximated by may term 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 3 +.... 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.)

12. Solve equation: N = N o e rt Alternative form: dN/dt = bN - dN = (b-d)N Rarely do b and d remain constant, but if well below what environment can support, then OK assumption. Each species has optimum environment with r = max

13. Figure 14.3

14. Figure 14.6

15. Human lice; r=.111/day How fast will a population that starts at 100 lice increase? (i.e., what is rate of increase of 100 lice?) dN/dt = rN =.111 x 100 = 11.1 lice/day

16. Human population in 1993 = 5,600,000,000 b = 26/1000, d = 9/1000 How fast was the population growing? dN/dt = rN = (.017)(5,600,000,000) = 95,200,000 (i.e., in excess of 1/3 US population per year)

17. Humans currently have b and d of 26 and 9 per 1000. How many years to double the population? N = N o e rt = N o x2 2 = e rt ln2/.017 = 40.77 yrs

18. 1700-1800 Human population from 600,000,000 ==> 900,000,000. Calculate r. r = ln (N/No) / t = ln(9/6)/100 =.0040547

19. Logistic Growth There has to be a limit. Postulate 2. Therefore add a second parameter to equation. dN/dt = rN + cN 2 call c = -r/K dN/dt = rN ((K-N)/K) N t = K/[1+((K-N o )/N o )e -rt ]

20. Figure 14.18

21. Figure 14.16

22. Optimal yield problem. dN/dt = rN - rN 2 /K d2N/dt2 = r - 2rN/K set = 0  N = K/2 If want maximum yield, should exercise continual cropping around N = K/2

23. Figure 14.17

24. Data ??

25. 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.

26. 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

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

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

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

30. Finite difference equations and Chaos N t+1 = aN t (1-N t ) Models populations with discrete, nonoverlapping generations, like many temperate zone insects. if 1<a<3, population settles to a steady state. if 3<a<3.57.., population settles into a stable cycle. if 3.57..<a<4, population apparently random or chaotic. if 4< a, N runs away to minus infinity.

31. This weird range of behaviors is generic to most difference equations that describe a population with a propensity to increase at low values and to decrease at high values. Similar behavior arises if there are many discrete but overlapping generations.