1. SPS Lecture: Dynamics of Population Growth Exponential Growth and the Fibonacci Sequence – solutions of simple linear differential and recurrence equations Plant Growth – logistic models for resource constrained growth Predator-Prey Dynamics – empirical data, phase plane model, and periodicity Salmon Dynamics – nonlinear phenomena and its utility

2. Thomas Robert Malthus Who was he ? What did he write ? http://www.blupete.com/Literature/Biographies/Philosophy/Malthus.htm #Intro Born south of London in 1766, one of eight children, he was privately taught by tutors and his father, who corresponded with Voltaire, Rousseau and Hume, and entered Cambridge at 18 where he studied theology and took up parochial duties in the Church of England. He married at the age of 38 and had 3 children. In 1805, he was appointed professor of Political Economy at the college at Haileybury, a college run by and for the general education of civil servants of the East India Company. Dies in 1834, the year slavery is abolished, outside of Bath. An Essay on the Principle of Population as it affects the Future Improvement on Society, with Remarks on the Speculation of Mr Godwin, M. Condorcet and other Writers (1798)

3. Thomas Robert Malthus What was the basis of his doctrine ? http://www.age-of-the-sage.org/philosophy/malthus.html "Population, when unchecked, increases in a geometrical ratio. Subsistence only increases in an arithmetical ratio. A slight acquaintance with numbers will show the immensity of the first power compared to the second".

4. n Malthusian Problem t = 1:100; plot(t,1.9+.1*t,t,1.025.^(t+19)) grid title(‘Malthusian Problem’) n Mathusian Crisis t = 1:200; plot(t,1.9+.1*t,t,1.025.^(t+19)) grid title(‘Malthusian Crisis’) Malthusian Problem and Crisis (MATLAB)

5. Charles Robert Darwin and Alfred Russel Wallace What are some of their major writings relevant to the life sciences ? http://www.age-of-the- sage.org/philosophy/wallace.html#Alfred_Russel_Wallace Malay Archipelago (1869), Contributions to the Theory of Natural Selection (1870) The Geographical Distribution of Animals (1876) Man's Place in the Universe (1903) http://www.literature.org/authors/darwin-charles/ The Voyage of the Beagle (1839) The Origin of Species by Means of Natural Selection (1859) The Descent of Man and Selection in Relation to Sex (1871)

6. Charles Robert Darwin and Alfred Russel Wallace How were they influenced by Thomas Malthus ? http://www.ucmp.berkeley.edu/history/malthus.html "In October 1838, that is, fifteen months after I had begun my systematic inquiry, I happened to read for amusement Malthus on Population, and being well prepared to appreciate the struggle for existence which everywhere goes on from long- continued observation of the habits of animals and plants, it at once struck me that under these circumstances favourable variations would tend to be preserved, and unfavourable ones to be destroyed. The results of this would be the formation of a new species. Here, then I had at last got a theory by which to work". Charles Darwin’s autobiography (1876) http://www.fact-index.com/a/al/alfred_russel_wallace.html His studies, eventually published as The Malay Archipelago, included the key influence of Thomas Malthus’ Essay on the Principle of Population, and led to his independent arrival at a theory of evolution similar to Charles Darwin’s.

7. Arithmetic and Geometric Sequences The preceding Malthusian Problem MATLAB Code generated an index sequence t = 1, 2, 3, …,201; an arithmetic sequence 1.9 +.1*t = 2, 2.1, 2.2, 2.3, …,10; and a geometric sequence (1.025)^20, (1.025)^21,...,(1.025)^100 and plotted both sequences versus the index sequence The n-th term of the arithmetic sequence can be expressed as a(n) = 1.9 +.1n The n-th term of the geometric sequence can be expressed as g(n) = (1.025)^(n+19)

8. Recursive Description The arithmetic sequence can be also be described by its first term together with the relationship of the (n+1)-st term to the n-th term for all positive integers n a(1) = 2 The geometric sequence can also be described in a similar way g(1) = (1.025)^(20) ~ 1.6386 a(n+1) = a(n) +.1 g(n+1) = (1.025)g(n) This way of describing a sequence can be used in a recursive computer program to compute any element of the sequence

9. n Computation of a a(1) = 2; for n = 1:99 a(n+1) = a(n)+.1; end plot(a) grid title(‘arithmetic seq a’) Recursive Computation of Sequences a and g n Computation of g g(1) = (1.025)^20; for n = 1:99 g(n+1) = 1.025*g(n); end plot(g) grid title(‘geometic seq g’)

10. Fibonacci Sequence Leonardo of Pisa (1170-1250), commonly called Fibonacci, proposed the following problem: A pair of rabbits is bought into a confined place. This pair and every other pair, begets one new pair in a month, starting in their second year. How many pairs will there be after one, two, …, months, assuming that no deaths occur ? f(1) = 1 Recursive Description Direct Solution 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … f(2) = 1 f(n+2) = f(n+1)+f(n)

11. General Fibonacci Sequences Result: The Fibonacci sequence is the sum of two geometric sequences To show this we exploit the amazing power of abstraction and consider the problem of describing ALL sequences that satisfy the equation If s is also a geometric sequence then there exists a number R such that We first look for geometric sequences that satisfy this equation R^2 = R + 1 s(n+2) = s(n+1) + s(n) s(n+1) = R s(n) therefore, combining the two equations above shows that R satisfies thereforeR 1 =.5(1+sqrt(5)) or R 2 =.5(1- sqrt(5)) hence we obtainf(n) = a (R 1 )^n - b (R 2 )^n where a = b =.2 sqrt(5) are determined to make f(1) = f(2) = 1

12. Tutorial Problems 1.How were Malthus, Darwin, and Wallace received in the 19 th century ? How did 20 th century ideologies claim that Darwinian theory supported their policies ? Are evolution and human biology still controversial subjects in the 21 st century ? 5. Plot (manually using graph paper) the first 7 values of the Fibonacci sequence. How well is it approximated by.2 sqrt(5) (R 1 )^n ? R 1 is called the Golden Ratio. 2. (Optional) Generate MATLAB plots where the ratio 1.025 is replaced by 1.01. Does the Malthusian crises eventually occur? 3. Justify why the values of R are what they are and show that the sequence f defined satisfies f(1) = f(2) = 1 4. (Optional) Derive a closed expression for f(n) that does not involve sqrt(5) by using the binomial theorem.

13. Golden Ratio It appears in natures angles, e.g. plantains, pinecones, peacocks and snails, and Nautilus: Hahn, p. 175, 454, 455, 456. Its remarkable number theoretic properties give it optimal circle partitioning properties, Lawton 1.

18. Growth Processes of a population of immortal cells that divide at the end of each calendar year is described by iterating a linear transformation # cells at least one year old# cells Fibbonaci sequence, whose ratio of successive terms approaches the golden ratio http://www.mcs.surrey.ac.uk/Personal/R.Knott/ Fibonacci/fib.html

19. Continuous Growth and Exponential Function of a large population is described by a differential equation where x = x(t) is size of population as a function of time t and is the specific rate of growth, and the solution for constant r is a function whose sequence of samples is a geometric sequence

20. Plant Growth has mass x whose specific growth rate is proportional to substrate S so if and as t increases x(t) approaches F = initial amount of substrate then calculus shows that

21. n Plant Growth F = 1; x0 =.2; k = 1; x = (x0*F)./(x0+(F-x0)*exp(-k*F*t)); t = 0:.01:10; plot(t,x) grid title(‘Plant Growth’) Plant Growth (MATLAB)

22. Predator-Prey Dynamics is described by the Lotka-Volterra equations Prey (herbivores) Solutions move counterclockwise along curves described by equations with constant positive coefficients. The equilibrium point is Predator (carnivores)

23. Nonlinear Salmon Dynamics Population grows from each generation to the next by MATLAB Simulations show dependence on parameters repro./survival parameter cannibalism parameter a = [5 10 11 15]; b = 2.3026; for j = 1:4 x(1)=1; for n = 2:40 x(n)=a(j)*x(n-1)*exp(-b*x(n-1)); end % n-loop figure(j) plot(x) grid end % j-loop

24. Tutorial Problems 6. (Optional) Use matrix tricks to derive a fast formula for the n-th term f(n) of the Fibonacci sequence. 7. A large population is doubling every year. What is the ratio that it increases by every month? 8. Assuming that food increases arithmetically, draw a plot that shows the approximate growth of population that is limited by available food. One unit of food production should be sufficient to sustain one population unit. Hint: study the plant growth plot. 9. (Optional) Critique Malthus’ simplistic arithmetic model and examine refinements Cobb-Douglas Production Functions. 10. Without cannibalism show that b = 0. Discuss why this model is unstable and leads to population explosion or decay.

25. References Hahn, Werner, Symmetry as a Developmental Principle in Nature and Art, World Scientific, Singapore, 1998 Lawton, Wayne, Kronecker’s theorem and rational approximation of algebraic numbers, The Fibonacci Quarterly, volume 21, number 2, pages 143-146, May 1983 Thompson, D’Arcy Wentworth, Growth and Form, Vol. I and II, Cambridge University Press, Cambridge, 1952 Gibbons, Michael Mesterton, A Concrete Approach to Mathematical Modelling,Wiley, Singapore 1995 Lotka, Alfred J., Elements of Mathematical Biology, Dover, NY, 1956 Ricker, W. E., Stock and recruitment, Candadian Journal of Fisheries and Aquatic Sciences 11, 559-623, 1954 Clark, Colin W., Mathematical Biosciences: The Optimal Management of Renewable Resources, Wiley, NY