Population is often recorded in a form of data set. Population of Normal, Illinois

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1 Population is often recorded in a form of data set Population of Normal, Illinois 1

2 Population of Venezuela 2

3 Population of world up to

4 Population of world 4

5 Population of world (carton) 5

6 Population in laboratory: bacteria 6

7 Experiment data of yeast cells of G.F. Gause (1934) 7

8 Questions: Find mathematical rules governing the growth Set up mathematical models Analyze the models Fit the model to the original data Model number 1 Let N n be the n-th generation of population N n+1 = N n + bn n dn n = (1 + d b)n n = λn n b =per capita reproduction rate, d =per capita mortality rate Malthus Model Solution: N n = N 0 λ n exponential growth when λ > 1, exponential decay when 0 < λ < 1 8

9 First Matlab program: plotting the time-number data (a sequence) A sequence is a function with domain being positive integers, and we can use a 1, a 2, a 3,, a n or {a n } to represent it The sequence can be generated by a recursive formula (or a difference equation a n+1 a n = f(a n )) a n+1 = f(a n ) and an initial condition a 0 = A 0. Biological question: for an observed data set, can we find a formula for the population sequence? 9

10 Example Wildebeest population growth. The wildebeest (or gnu) is a dominant species in the Serengeti. The following data of wildebeest abundance was collected by the Serengeti Research Institute: year population (in thousands) Question: assuming N n+1 = λn n can be used to model the data. How do we find N 0 and λ? 10

11 Data fitting for Malthus model: (estimate λ and N 0 ) (a) Get a set of data: (t 1, P 1 ), (t 2, P 2 ),, (t n, P n ). (b) Take ln to P i, let Q i = ln(p i ), and get new data set (t 1, Q 1 ), (t 2, Q 2 ),, (t n, Q n ). (c) Put the data set (t i, Q i ) to your linear regression program and get the output slope k and intercept b. (d) In the solution of Malthus model: N n = N 0 λ n, we have ln N n = ln N 0 + n ln λ, so b = ln N 0, k = ln λ. Then N 0 = e b and λ = e k. So an exponential function which best fits the data is N n = e b+kn, where k and b are found in linear regression 11

12 Linear Regression: The goal of linear regression is to adjust the values of slope and intercept to find the line that best predicts Y from X. More precisely, the goal of regression is to minimize the sum of the squares of the vertical distances of the points from the line. Method of linear regression: (a) get a set of data: (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ). (b) put it into a computer linear regression program Mechanism of linear regression: Data: (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ). Goal: Find a line y = kx + b minimize the sum of the squares of the vertical distances of the points from the line. Vertical distance from (x i, y i ) to y = kx + b: kx i + b y i 12

13 Mathematical problem: Find k and b which minimize f k = n i=1 f(k, b) = n i=1 2(kx i + b y i )x i = 0, f b = So solve k and b from n x i i=1 n x 2 i i=1 k + nb = k + n i=1 n x i i=1 y i and b = n i=1 (kx i + b y i ) 2 x i y i. n i=1 2(kx i + b y i ) = 0. 13

14 Linear difference equation: a n+1 = λa n, what can happen? Solution: a n = a 0 λ n, and a n = 0 is an equilibrium. Qualitative behavior? λ = 0, 1, 1 λ = 0.5, 0.5, 2, 2 Inhomogeneous linear difference equation x n+1 = ax n + b (see homework) 14

15 Nonlinear model of population N n+1 = λn n S(N n ) S(N) is the survival rate of the population which depends on the size of population, and λs(n) is the per capita reproduction rate This is a density dependent population growth. Usually unlimited growth is unrealistic, as population grow the amount of resources available per individual decreases. How to determine S(N)? 1. From certain assumptions to derive the mathematical forms 2. Available data suggests the shape of function, then find functions with such shape. 15

16 The relationship between number of survivors and density for four stored product beetles. After Bellows, T. S The Descriptive Properties of Some Models for Density Dependence. Journal of Animal Ecology, Vol. 50, No. 1. pp

17 N n+1 = λn n S(N n ) Bellow s model: NS(N) = N 1 + (an) b, a > 0, b 1. A similar model: Hassel s model NS(N) = N (1 + an) b, a > 0, b > 0. Parasitoid fly b = 0.5 and λ = 3.2; Bug: Saccarosydne saccharivora b = 0.4 and λ = 13.5 Mosquito β = 1.9 and λ = 10.6 Potato beetle b = 3.4 and λ = 75.0 Blowfly b = 10 and λ 100 Hassell, M.P., Lawton, J.N. & May, R.M. (1976) Patterns of dynamical behaviour in single-species populations. Journal of Animal Ecology, 45,

18 Simplify the equation: (nondimensionalization) Bellow s equation: N n+1 = Let x n = an n. Then x n = λn n 1+(aN n ) b = N nf (N n ) = f(n n ).. λx n 1+x b n (The number of parameters reduces from 3 to 2.) Hypothesis: insect populations are regulated by intraspecific (within-species) competition for some resource in short supply. (growth rate per capita F (N) is decreasing, compensatory) under-compensation: if 0 < b < 1, lim N f(n) = over-compensation: if b > 1, lim N f(n) = 0 exact-compensatory: if b = 1, lim N f(n) = C > 0 If F (N) is increasing for some N, then the system exhibits depensation (Allee effect) 18

19 Now what to do? Let s do experiment using computer... x n+1 = λx n 1 + x b. n plot the sequence (time series) plot cobweb diagram more mathematics coming... 19