1.2. Introduction to Modeling

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1 G. NAGY ODE August 30, Section Objective(s): Population Models Unlimited Resources Limited Resources Interacting Species 1.2. Introduction to Modeling Population Model with Unlimited Resources. Remarks: In the real world food is never unlimited, so unlimited food models are useless. However, bacteria in a petri dish have unlimited food for a while; so the unlimited food model is useful for a while. Simple models may be good approximations to complex situations for a certain time. More complicated models are often constructed from simpler ones. Case 1: The Unlimited Resources Model. (Review) Problem: Describe a bacteria population when the have unlimited food, when the population rate change per capita is r > 0. P (t) = r P (t), r > 0. P (t) = P (0) e rt.

2 2 G. NAGY ODE august 30, 2018 Case 2: The Unlimited Resources with Immigration Model. Problem: Describe a village population when they have unlimited food, the rate of population growth per capita is 3, and they have an immigration rate of 9 persons per unit time. P (t) = 3 P (t) + 9, P (t) 3 P (t) + 9 = 1! P! (t) 3 P (t) + 9 dt = dt! dp 3p + 9 dt = t + c 0 1! 3 dp p + 3 dt = t + c 0 ln p + 3 = 3t + c 1 P (t) + 3 = e 3t+c1 = e 3t e c1 P (t) + 3 = c 2 e 3t P (t) = c 2 e 3t 3. P (0) = c 2 3 c 2 = P (0) + 3 P (t) = (P (0) + 3) e 3t 3.

3 G. NAGY ODE August 30, Case 3: Radioactive Decay. Problem: Describe the amount of radioactive material with half-life τ. N (t) = r N(t), r > 0. Theorem: kτ = ln(2). N(t) = N(0) e rt. Question: What happens to the half-life when the radioactive constant r is large? Read in Lecture Notes: Using radioactive decay to date remains.

4 4 G. NAGY ODE august 30, Population Models with Finite Resources. If the population P (t) is small : P (t) r P (t) > 0. If the population P (t) is large : P (t) < 0. Definition 1. The logistic equation for the function P, which depends on the independent variable t, is P (t) = r P (t) 1 P (t) #, (1.2.1) r > 0 is the growth constant and > 0 is the carrying capacity. Example 1.2.1: Suppose the function P is solution to the logistic equation P (t) = r P (t) 1 P (t) #. (a) For what values of P is the population in equilibrium that is, time independent? (b) For what values of P is the population increasing in time? (c) For what values of P is the population decreasing in time? Solution: (a) If P is an equilibrium solution, then onstant, so, P = 0. The equation says 0 = P = r P 1 P # P = 0 or P = Pc. (b) If P is increasing, then P > 0, then implies (c) If P is decreasing, then P < 0, but P (t) = r P (t) 1 P (t) # > 0, r > 0, P! 0 1 P (t) # > 0 0 < P (t) <. 1 P (t) # < 0 P (t) >, since annot be negative. Discuss the meaning of the carrying capacity.

5 G. NAGY ODE August 30, Interacting Species Model. Problem: Write a simple model to describe how rabbits and sheep populations evolve in time when they compete on the grass on a particular piece of land. Solution: Suppose the species do not interact. R are rabbits and S are sheep. R = r r R 1 R # R c S = r s S 1 S #, S c where r r, r s are the growth rates and R c, S c are the carrying capacities. Introduce the effect or sheep on rabbits. R = r r R 1 R # c 1 R S, c 1 > 0. R c The product measures the encounters on the field. Introduce the effect of or rabbits on sheep. S = r s S 1 S # c 2 R S, c 2 > 0. S c The product measures the encounters on the field.

6 6 G. NAGY ODE august 30, 2018 Definition 2. The interacting species equation for the functions x and y, which depend on the independent variable t, are x = r x x 1 x # + α x y x c y = r y y 1 y # + β x y, y c where the constants r x, r y and x c, x c are positive and α, β are real numbers. Example 1.2.2: The following systems are models of the populations of pair of species that either compete for resources (an increase in one species decreases the growth rate in the other) or cooperate (an increase in one species increases the growth rate in the other). For each of the following systems identify the independent and dependent variables, the parameters, such as growth rates, carrying capacities, measure of interactions between species. Do the species compete of cooperate? (a) (b) dx dt = c x 2 dx 1x c 1 b 1 xy K 1 dt = x x xy Solution: (a) dy dt = c y 2 2y c 2 b 2 xy. K 2 (b) dy dt = 2 y y xy. The species compete. t is the independent variable. x, y are the dependent variables. c 1, c 2 are the growth rates. K 1, K 2 are the carrying capacities. B 1, b 2 are the competition coefficients. The species coperate. t is the independent variable. x, y are the dependent variables. 1, 2 are the growth rates. 5, 12 (not 6) are the carrying capacities. 5, 2 are the competition coefficients. Question: If x are elephants and y are chipmunks, then is b 1 > b 2 or b 2 > b 1? Answer: b 2 > b 1.