x'=r (1 - 2 / 2 x where r and K are positive constants, is called...
#2 Consider the following model for the dynamics of a population of size N (measured as number of individuals x 10) over time (in months) that is subject to harvesting: The population grows according to a logistic equation in the absence of harvesting and h is a constant per a) Find all equilibria and determine the values of h for which each is stable or unstable. 4aestng andcnstant capita harvest rate. b) Construct the bifurcation plot: plot the equilibria from...
4. Consider the equation N,+1 = N, exp[r(1-N/K)] This equation is sometimes called an analog of the logistic differential equation (May, 1975). The equation models a single-species population growing in an environment that has a carrying capacity K. By this we mean that the environ- ment can only sustain a maximal population level N = K. The expression reflects a density dependence in the reproductive rate. To verify this observa- tion, consider the following steps: (a) Sketch A as a...
Suppose that a population that evolves according to the logistic growth is harvested at the constant rate H. Then the population size (t) satisfies the equation INNK-NU where the new term -H on the right-hand side accounts for the harvesting, r> 0 is constant, K is the carrying capacity and H is a constant greater than or equal to 0. (a) (1 mark) First suppose that there is no harvesting, that is, H = 0. Let r = 0.3 and...
Please help with part c of ii - v
2 Snow Crab Fishing Suppose the snow crab population in the Gulf of Saint-Lawrence grows according to an instanta- neous logistic growth that is given by, F(x) X1- where X is the biomass (ie. quantity, in tons) of snow crabs, r = 0.1 is the intrinsic instantaneous (a) What are the two biomass levels X that define the two biological equilibria of the snow crab (b) What is the biomass that...
Consider the dynamics of a directly transmitted viral microparasite to be modelled by the system 1 d X dY where b, B and r are positive constants and X, Y and Z are the number of suscep- tibles, infectives and immune populations respectively. Here the population is kept constant by births and deaths (with a contribution from each class) balancing. Show that there is a threshold population size, Ne, such that if N < Nc = (b + r) the...
Exercises 1. Verify equation (3) 2. Use the techniques of Section 13.7 and the fact that P(0) = 10 to solve equation (5). 3. The carrying capacity of Atlantic harp seals has been estimated to be C = 10 million seals. Let 1 = 0 correspond to the year 1980 when this seal population was estimated to be about 2 mil- lion. (Data from: Fisheries and Oceans Canada.) (a) Use a logistic growth model = kP(C - P) with k...
23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity a. Find all of the critical p there are no critical points i and two critical points if a O b. Draw the phase line each critical point is asy Consider...
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...