Exercise:
A very basic population model for the coexistence of foxes and rabbits is defined by the differential s fractextrmdRtextrmdt a R - c F fractextrmdFtextrmdt -b F + d R where all the constants a b c d are positive numbers. abcliste abc Discuss the asptions of this model in words i.e. discuss the development of the fox and rabbit population if they were left to themselves and how one population affects the other one. abc What are the conditions for the parameters for oscillatory solutions i.e. a system corresponding to a centre? abc What are the conditions for the parameters for solutions where both populations first decrease and then explode? abcliste

Solution:
abcliste abc Without foxes the rabbit population would be given by dot R aR The rabbit population would grow exponentially due to the lack of an enemy. Obviously this would only work in an environment with unlimited food resources. vspace.cm The fox population in a world without rabbits would be governed by dot F -b F The fox population would decrease exponentially due to the lack of prey rabbits. The cross terms can be understood as follows: dot R -c F describes the effect of the fox population on the rabbit population. The larger the number of foxes the faster the number of rabbits decreases. dot F d R describes the effect of the rabbit population on the fox population. The more rabbits the faster the number of foxes increases. abc The system is defined by the matrix bf A pmatrixa & -c d & -b pmatrix The trace and determinant of this matrix are tau a-b Delta a-b--cd cd-ab It follows for the eigenvalues lambda_ fractaupmsqrttau^-Delta fraca-bpmsqrta-b^-cd-ab fraca-bpmsqrta+b^-cd For a centre the eigenvalues have to be purely imaginary. As a first condition it follows that a b The condition for the square root to be imaginary is that cd & ab a^ As a numerical example we choose the following values: a b c d The eigenvalues are lambda_ frac-pmsqrt+^- pm i The diagram below shows the direction field and some orbits phase portrait. center includegraphicswidthtextwidth#image_path:rabbits-foxes-centr# center Remark: Since negativ populations obviously do not make sense we have to erpret the values as deviations from the respective equilibrium populations. Alternatively we could transform the homogeneous system o an inhomogeneous system with fixed po R^* F^* equilibrium population. For the transformation see exercise urlhttps://texercises.com/exercise/inhomogeneous-linear-systems/?colsystems-of-linear-differential-s. abc The case where the populations first t to the fixed po and then grow beyond any limits corresponds to a saddle. The condition for a saddle is that one eigenvalue is a real negative number stable and the other eigenvalue a real positive number unstable. With lambda_ fractaupmsqrttau^-Delta this is the case if either tauneq and Delta. It follows that a &neq b and cd & ab As a numerical example we choose the following values: a b c d The eigenvalues are then lambda_ frac-pmsqrt-^-left - right frac-pmsqrt so lambda_ and lambda_. The diagram below shows the direction field and some orbits phase portrait. vspace.cm center includegraphicswidthtextwidth#image_path:rabbits-foxes-saddle# center abcliste