MATA35H3 Final: summary 4.2

The population of an endangered hawk species and its natural prey the field mouse, originally 30 hawks and 2000 mice, is represented by the following model: h k+1 = 0.55hk + 0.005mk mk + 1 = -18 hk + 1.2mk where hk and mk represent the hawk and mice population respectively in year k. Make sure you understand what each parameter means. Express the model in the matrix form V k + 1 = AVk, and write V0 = Find the eigenvalues and corresponding eigenvectors of A, use them to diagonalize the matrix A, and write a general formula to calculate Vk. Calculate the predicted population every few years for the next 30 years or so and, using eigenvalues, reason whether an equilibrium will be reached. In 5 years, biologists try to improve the hawk population by introducing 1000 mice in the hawk hunting area. Predict and compare the new hawk population 10 years from now. From that point on, the hawk survival rate drops to 0.50. The biologists are alarmed that this may wipe out the hawk population. Test to see what the model is predicting and explain what is happening. Describe the trajectories of the dynamical system with matrix A = [0 1 -1 0].

Fixed. If beta denotes the juvenile survival rate, show that the population: stabilizes if beta = 1-alpha / m; becomes extinct if beta 1-alpha / m. The population of an endangered hawk species and its natural prey the field mouse, originally 30 hawks and 2000 mice, is represented by the following model: h k+1 = 0.55 hk + 0.005mk m k+1 = -18 hk + 1.2 m k where hk and mk represent the hawk and mice population respectively in year k. Make sure you understand what each parameter means. Express the model in the matrix form V k+1 = AV k, and write V0 = [30 2000]. Find the eigenvalues and corresponding eigenvectors of A, use them to diagonalize the matrix A, and write a general formula to calculate Vk. Calculate the predicted population every few years for the next 30 years or so and, using eigenvalues, reason whether an equilibrium will be reached. In 5 years, biologists try to improve the hawk population by introducing 1000 mice in the hawk hunting area. Predict and compare the new hawk population 10 years from now. From that point on, the hawk survival rate drops to 0 50 The biologists are alarmed that this may wipe out the hawk population. Test to see what the model is predicting and explain what is happening. the trajectories of the dynamical system with matrix

(1 point) The population P of some species in an environment with limited resources is usually modelled by the logistic function 1 be-ct where a, b, c are positive constants and t represents time. It is known that the initial population is 155 and increases to 196 after 2 years. After a long time, the population will approach to the carrying capacity of 1070. (a) Find a, b and c so that the logistic equation P can be used to model the population of this species after t years. Round your answers to at least 5 significant figures. a= (b) What is the initial growth rate? Round your answer to at least 3 significant figures. Growth rate = /year (c) How long does it take for the population to reach 62% of the carrying capacity? Round your answer to at least 3 significant figures. It takes years.