Population Growth Modeling: Malthus Model and Linear Regression, Lab Reports of Mathematics

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Population is often recorded in a form of data set

Population of Normal, Illinois

Population of Venezuela

Population of world

Population of world (carton)

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

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 Nn be the n-th generation of population

Nn+1 = Nn + bNn − dNn = (1 + d − b)Nn = λNn b =per capita reproduction rate, d =per capita mortality rate

Malthus Model Solution: Nn = N 0 λn exponential growth when λ > 1, exponential decay when 0 < λ < 1

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 1961 1963 1965 1967 1971 1972 1977 1978 population (in thousands) 263 357 439 483 693 773 1444 1249

Question: assuming Nn+1 = λNn can be used to model the data.

How do we find N 0 and λ?

Data fitting for Malthus model: (estimate λ and N 0 )

(a) Get a set of data: (t 1 , P 1 ), (t 2 , P 2 ), · · · , (tn, Pn).

(b) Take ln to Pi, let Qi = ln(Pi), and get new data set (t 1 , Q 1 ),

(t 2 , Q 2 ), · · · , (tn, Qn).

(c) Put the data set (ti, Qi) to your linear regression program

and get the output slope k and intercept b.

(d) In the solution of Malthus model: Nn = N 0 λn, we have

ln Nn = ln N 0 + n ln λ, so b = ln N 0 , k = ln λ. Then N 0 = eb^ and

λ = ek. So an exponential function which best fits the data is

Nn = eb+kn, where k and b are found in linear regression

Mathematical problem: Find k and b which minimize

f (k, b) =

∑^ n

i=

(kxi + b − yi)^2

∂f ∂k

∑^ n

i=

2(kxi + b − yi)xi = 0,

∂f ∂b

∑^ n

i=

2(kxi + b − yi) = 0.

So solve k and b from

 ∑^ n

i=

xi

  (^) k + nb = ∑^ n

i=

yi and



 ∑^ n

i=

x^2 i

  (^) k +

  ∑^ n

i=

xi

  (^) b = ∑^ n

i=

xiyi.

Linear difference equation: an+1 = λan, what can happen?

Solution: an = a 0 λn, and an = 0 is an equilibrium. Qualitative

behavior?

λ = 0, 1 , − 1

λ = 0. 5 , − 0. 5 , 2 , − 2

Inhomogeneous linear difference equation xn+1 = axn + b

(see homework)

The relationship between number of survivors and density for four stored product beetles. After Bellows, T. S. 1981. The Descriptive Properties of Some Models for Density Dependence. Journal of Animal Ecology, Vol. 50, No. 1. pp. 139-156.

Nn+1 = λNnS(Nn)

Bellow’s model: N S(N ) =

N

1 + (aN )b

, a > 0, b ≥ 1.

A similar model: Hassel’s model N S(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, 471-486.

Now what to do? Let’s do experiment using computer ......

xn+1 =

λxn 1 + xbn

plot the sequence (time series)

plot cobweb diagram

more mathematics coming........