Consider the survival of a population of species S. Assume that if the population size fallbelow a minimum survival level m, then the species will become extinct. Assume also thathe population is limited by the carrying capacity M of the environment. That is, if thpopulation is above M, then it will experience a decline because the environment cannosustain that large population level.(a) Let an be the population of species S after n years. Explain briefly the modelan+1 = an+k(M-an) (an - m),where k > 0. Does it make sense in terms of the description above?(b) Find the fixed points of the model and determine their stability. Assume that M =5000, m= 100, and k = 0.0001.(c) Sketch the graphs of an versus n for various initial conditions. You may use spreadsheet. Indicate the necessary values used in each graph.(d) The model has two serious shortcomings. What are they? (Hint: Consider differeninitial values.)

Given: Survival of a population of species S. Population size fall below minnimum survival level m,…

Answered: Consider the survival of a population…

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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A model is built to explain the evolution of spending on tourism and recreation in a certain group of families (Y in USD/person/year). As potential explanatory variables are considered: X1 average annual income per person in the family (in USD), X2 – number of people in the family, X3 – nature of employment of the head of household If X3=1, when the head of the family is self-employed, X3=0, when the head of the family is an employee Based on the collected data, linear correlation coefficients between variables were calculated and obtained 0,84 R, = -0,47 [1 -0,55 0, 62 R= 1 -0,42 0,75 1 Using Hellwig's method, select the optimal combination of explanatory variables for the tourism and recreation expenditure model.
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A population of fish is living in an environment with limited resources. This environment can only support the population if it contains no more than M fish (otherwise some fish would starve due to an inadequate supply of food, etc.). There is considerable evidence to support the theory that, for some fish species, there is a minimum population m such that the species will become extinct if the size of the population falls below m. Such a population can be modelled using a modified logistic equation: dP =(1-)(-) m dt M
Please answer this question using the assumptions from the Solow Model In a country, the production function in terms of output per capita, y, and capital per labor, k, is y = k(12) The fraction of output saved, s, is 25% and the depreciation rate, δ, is 0.05. Assume that the population does not grow. What is the steady-state of capital per worker, k*, and output per worker, y*? What happens to the steady-state values if the population growth rate rises to 5%? By how much would the country have to increase its savings rate in b) in order to reach the same levels of steady-state values as in a)? Does the country’s growth rate increase in c) compared to a)? how?
number of people who will be infected by t = 12 days. 7. Assume we are considering the survival of whales and that if the number of whales falls below a minimum survival level m, the species will become extinct. Assume also that the population is limited by the carrying capacity M of the environment. That is, if the whale population is above M, then it will experience a decline because the environment cannot sustain that high a population level. a. Discuss the following model for the whale population dP dt = k(MP)(Pm) where P (t) denotes the whale population at time t and k is a positive constant. b. Graph dP/dt versus P and P versus t. Consider the cases in which the initial population P(0) = Po satisfies Po

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