The evolution of a population with constant migration rate M is described by theinitial value problemdP= kP + M.dtP(0) = Po.(a) Solve this initial value problem; assume k is constant.(b) Examine the solution P(t) and determine the relation between the constants kand M that will result in P() remaining constant in time and equal to Po- Explain,on physical grounds, why the two constants k and M must have opposite signs toachieve this constant equilibrium solution for P(t).

The initial value problem describing the evolution of a population with a constant migration rate M…

Answered: The evolution of a population with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve the initial value problem y = []» ym = [*.] -7 4 У, -6 7 у, у(0) — || -4
(a) Suppose we create a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2). Height is measured in cm, bacteria is measured in thousand per ml of soil (so 1000 is 1 unit), and type of sun = sun and type of sun = between shrubs with a 5500 bacteria and those with a 4000 bacteria count. Assuming the sun is the 0 if the plant is in partial 1 if the plant is in full sun. What will be the height difference on average same for both. (b) Let's say after a few training steps your model is in this state: Y = 33 + 4.6 * X1+8* X2. Explain what the effect of the plant being in sun vs. partial shade will have on the average height of the shrub if there are no bacteria in the soil, based on the model.
5. Company A plans to market a new product. The marketing department has determined that the demand for these products is p = - 0.05x + 620,0sx< 30,000, where p denotes the product's unit price (in dollars) and x denotes the quantity demanded. Also, the company estimates that the total cost (in dollars) incurred in manufacturing x products per year of production will be C(x) = 100x +710,000. Find the marginal revenue function and the marginal profit function.
An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation 1. = a² y_a² y at² Ox² (1) Equation 1 can be written in the form of Equation 2 using the central finite difference approximation of the second derivative of their respective variables. The subscripts represent locations, while superscripts represent time steps. y=By+Cy+ Dy₁₁ + Ey i+1 (2) +1 a) Express B, C, D and E in terms of A t and A x. Note that the coefficient of y, is equal to one. b) The string is discretised into five nodes equally distanced, with A t = 0.2. Write down equations as a result of applying Equation 2 to the domain. Subscripts must be replaced with correct node numbers, while superscripts can be retained not required to incorporate the boundary and initial conditions yet. c) Assemble the three equations above in the form of matrix {y}+1 = [A] {y}'+ {y}+{BC}, where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}+1, {y} and {y} are…
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Check all the properties of the logistics DDS that are true: 1) The logistics DDS is more realistic than the exponential model because it takes limitations of the environment into account. 2) The logistics DDS has a quadratic undating function. 3) The logistics DDS has 2 equilibrium values, the trivial one at zero and the non-trivial one at the carrying capacity. 4) The logistics DDS has unlimited growth because it is a power function 5) Both of the equilibrium values of the logistics DDS are stable.

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