Recall the Lotka volterra model of predator-prey interactions from the lectures: ()()=()()()=()()() where () is the number of prey, () is the number of predators, and are the birth and death rates of prey. and are the birth and death rates of predators. a) In this model, what do the dynamics of the prey population over time look like in the absence of predators ( (0)=0 )? Do you think this is reasonable? (A verbal description without maths is sufficient, although you are welcome to use equations if you like) (3 marks) A common alteration of the birth rate in Lotka-Volterra-style models, which somewhat deals with the issue raised in question a), is the following: ()=()(1()) where >1 is a parameter we will ask you to describe. b) Can you verbally describe how this deals with the issue of question a? What is the maximum value the prey population could take after starting from an initial population of 1 ? How would you interpret the parameter ? (3 marks) We are now going to deal with a model of species competition. Two species which don't predate on each other, but compete for a limited food source. Their respective populations are 1() and 2() . 1()2()=11()(11()11221)=21()(12()22112) Notice that the differential equation describing population change of each species , is exactly the same as for the differential equation of question b, except it includes an additional term . This term models inter-species competition. For instance, if 2() is large, this term exerts a negative pull on the population growth rate of 1() . We are now going to reduce the number of parameters in our model by nondimensionalisation. The nondimensionalised model is: d1d()d2d()=1()(11()122())=2()(12()211()) where the new parameters, expressed in terms of the original parameters, are: 1=11,2=22,=1,=?????12=122121=2112 c) Figure out what is, in terms of the parameters of the original equation. Describe what it represents biologically (Possible even if you couldn't find its formula using common sense and after running the simulations below) (2 marks) d) We're now going to simulate the differential equation several times. So write some code for simulating the differential equation, using scipy.integrate.solve_ivp. Of course, to actually simulate the system, you will need initial conditions and parameter values: (4 marks) For each simulation, we are going to have the following initial conditions: 1(0)=2(0)=0.1 However, we will change the parameters repeatedly, to see how they influence the behaviour of the model. Simulate repeatedly with each of the following sets of parameter values (no need to write or plot anything for the report). The point of this is only to get a feel for how the model behaviour changes with the parameters. 12=0.921=1.1=1.6 12=1.121=0.9=1.6 12=1.121=0.9=0.6 12=1.121=0.9=0.6 12=0.621=0.4=5 12=0.421=0.6=5 e) From simulating, with these parameter combinations, what seems to be more important for long term species survival in this model: its natural growth rate or the degree to which it is suppressed by its competitor? Plot ONE simulation from the parameter values above that supports your conclusion. (3 marks) f) Now calculate the four fixed points of the differential equation. You will find three are easier, and one is harder. For all four fixed points, briefly describe (1/2 sentences) what each fixed point represents in terms of the competition between the two species. Although you don't need the previous simulations for this question, they might be a useful sanity check for your answers! (5 marks) Our system of differential equations is in the general form: ()=(()), where () is the vector [1(),2()] . Note that is just time with rescaled units (like minutes vs seconds), so you can treat it as time. g) Calculate the Jacobian matrix (2 marks) ()=dd() h) Evaluate this Jacobian matrix at the two fixed points you arrive at when using the parameters: 12=0.921=1.1=1.6 12=1.121=0.9=1.6 What are the requirements on the values of 12 and 21 for these fixed points to be stable? (3 marks) To do this, figure out conditions on when the Jacobian matrix has appropriate eigenvalues. You don't necessarily need to calculate the individual eigenvalues. Instead note that The trace (sum of diagonal elements) of the matrix is equal to the sum of eigenvalues The determinant of the matrix is equal to the product of the eigenvalues ...and we only need to know the sum and product of two numbers (the eigenvalues) to see if they are both negative!