MATH 221 Biocalculus II Project 5 Nonlinear Systems of Difference Equations. BIO: To explore some modifications of the Nicholson-Bailey Model

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1 Goals MATH 221 Biocalculus II Project 5 Nonlinear Systems of Difference Equations MATH: To analyze nonlinear systems of difference equations BIO: To explore the Nicholson-Bailey Model BIO: To explore some modifications of the Nicholson-Bailey Model COMP: Implement nonlinear difference equations in EXCEL dynamically Computational Tools: EXCEL (Make sure the Data Analysis Add-In loaded.) and Maple as needed Create the Nicholson Bailey Spread Sheet The Nicholson Bailey Model is given by N t+1 = bn t e apt = f(n t,p t ) P t+1 = cn t (1 e apt )=g(n t,p t ), where N t is the size of the host population in the tth generation, P t is the size of the parasitoid population in the tth generation, b and c are growth parameters, and a is parameter that characterizes the fraction of the the host population that escapes parasitism. (Specifically, the factor e apt is the fraction of the N t that escapes parasitism.) We denote the initial populations of the host by N 0 and the parasitoid by P Label the current worksheet, Sheet1, Home. 2. Create Sliders for the parameters a, b, andc, and the initial population values N 0 and P 0 in the model. 3 Initial Number of Hosts N0 Slider from 0 to 20, incremental change 1, linked to D3 4 Initial Number of Parasitoids P0 Slider from 0 to 20, incremental change 1, linked to D4 5 Host Escape Rate a =E5/100 Slider from 0 to 100, incremental change 1, linked to E5 6 Host Growth Parameter b =E6/10 Slider from 0 to 100, incremental change 1, linked to E6 7 Parasitoid Growth Parameter c =E7/10 Slider from 0 to 100, incremental change 1, linked to E7 Note: Cells D3 and D4 will be filled with slider values. 3. Click on Sheet2, and change its name to Data. 4. Label cell A4 T, cell B4 N, cell C4 P. 5. Fill the A column from A5 to A205 with the integers 0, 1,2,..., Enter =Home!$D$3 in B5 and =Home!D4 in C5. 7. In B6, enter =Home!$D$6*B5*EXP(-Home!$D$5*C5) and in C6, enter =Home!$D$7*B5*(1 - EXP(-Home!$D$5*C5)) 8. Copy the formulas in B6 and C6 down their respective columns through row 205. At this point, the times series data has been generated. 9. In the Home worksheet, create a phase plane diagram of P versus N. Label the axes and give the graph a title. Use an XY scatter plot without connecting the points. (Does this figure seem helpful?) 10. By copying the phase plane diagram, create a new phase plane diagram in which the points are connected by smoothed lines. (Does this figure seem helpful?) 1

2 11. In the Home worksheet, create a chart containing plots of P versus T and N versus T. Make the horizontal axis range from 0 to 50 and the vertical axis range from 0 to 100. Again, use XY scatter plots and connect the points with smoothed lines. (Does this plot seem helpful?) 12. Play with your parameter sliders and look at this third plot to gain some insight into the behavior of this model. Will the populations approach equilibrium values? 13. Find the equilibria of the Nicholson Bailey Model. Let ( ˆN, ˆP ) denote the nontrivial equilibrium of the system. Enter the expression for ˆN in terms of the cells containing the corresponding parameter values into cell L4 and the expression for ˆP into cell L5. Label these cells by entering appropriate titles in cells H4 and H5, respectively. 14. We need to compute the Jacobian of this system evaluated at and its eigenvalues to determine whether the nontrivial equilibrium is locally stable or unstable. Calculuate the matrix: J(N,P) = [ F N G N F P G P ] Now evaluate this matrix at ( ˆN, ˆP ). What conditions are on the parameter b so that ˆN is a positive quantity? What is the biological significance of this nontrivial equilibrium. Explain. Add the nontrivial equilibrium point ot the phase plane plots. Enter the Jacobian matrix in terms of the cells corresponding to appropriate parameter values and into cells B12, C12, B13, and C13. (You will either need to calculuate the Jacobian by hand or using a computer algebra system. 15. For an equilibrium point of system of two nonlinear difference equations, if the magnitudes of both eigenvalues of the Jacobian matrix are less than one, the equilibrium is locally stable. If at least one eigenvalue has magnitude greater than 1, the equilibrium is unstable. The spreadsheet will determine stability by directly calculating the eigevalues and by using the Jury Conditions below. We begin this process by placing the determinant of the Jacobian in cell F12 and the trace of the Jacobian in cell F13. Name cell F12 Det and cell F13 Trace. 16. Excel does not compute in terms of complex numbers, but we need to be able to work with complex eigenvalues. To do this, we perform the following calculations that return the real and imaginary parts to solutions of quadratic equations separately. Enter the following in K21: =SQRT((Trace^2-4*Det+ABS(Trace^2-4*Det))/2) Enter the following in K22: =SQRT((-(Trace^2-4*Det)+ABS(Trace^2-4*Det))/2) Name K21 alpha and K22 beta. 17. Enter Lambda1 and Lambda2 into H12 and H13, respectively. Make sure the Data Analysis Add-In has been installed so that the following commands will work. Enter the value of the eigenvalue Lambda1 in cell I12: =COMPLEX(ROUND((Trace+alpha)/2,4),ROUND(beta/2,4)) Enter the value of eigenvalue Lambda2 in cell I13: =COMPLEX(ROUND((Trace-alpha)/2,4),ROUND(-beta/2,4)) 2

3 Enter the magnitude of these eigenvalues in cells J12 and J13. The formula for the norm of Lamdba1 is given by: =IMABS(COMPLEX((Trace+alpha)/2,beta/2)) By further playing with several different parameter values, what do you predict about the stability of the nontrivial equilibrium? 18. (Jury Conditions.) For a system of two nonlinear difference equations, there are necessary and sufficient conditions that indicate when both eigenvalues of the Jacobian are less than one in magnitude. The conditions are the following: If the characteristic equation for J = J( ˆN, ˆP )is λ 2 Trace(J)λ +Det(J) =0, then both roots of the equation (the eigenvalues) will have magnitude less than one if and only if 2 > 1+Det(J) > Trace(J). Implement this test in Excel as follows: Enter 1 + Det(J) in N6, Trace(J), in N7, =1 + Det in O6, and =ABS(Trace) in O7. Enter Jury Test in N9, Stability in N10, =IF(AND(2 > $O$6,$O$6> $O$7), SATISFIED, FAILED ) in O9, and =IF($O$9= SATISFIED, LOCALLY STABLE, UNSTABLE ) in O10. What you observe here should be consistent with the observed magnitudes of the eigenvalues of the Jacobian. 19. Calculate Trace(J) anddet(j) in general and show that Det(J) is greater than 1 for the values of b satisfying the condition above. (Note that you can accomplish this by showing that the function S(b) =b 1 b ln(b) < 0, on the appropriate domain for b. WhatisS(1) and what is S (b)?) Draw a conclusion about the stability of ( ˆN, ˆP ). 20. Determine the stability of the trivial equilibrium as well. 21. Use the spreadsheet to observe how the Nicholson Bailey Model behaves with the following parameter choices. Describe the dynamical behavior in biological terms in each case. Print each time series graph. (a) N 0 = 15, P 0 =8,a =0.02, b =1.5, c =1.5 (b) N 0 = 15, P 0 =0,a =0.02, b =1.5, c =1.5 (c) N 0 = 15, P 0 =8,a =0.02, b =0.5, c =1.5 (d) N 0 = 15, P 0 =8,a =0.068, b =2,c =1 Negative Binomial Model With a few minor changes, we can implement the spreadsheet for the Negative Binomial model: ( N t+1 = bn t 1+ ap ) k = f(n t,p t ) k ( P t+1 = cn t (1 1+ ap ) ) k = g(n t,p t ), k where all the parameters are the same as in the previous model, with the addition of the binomial power k. In this model, ( ) 1+ ap k k serves as the fraction of the host populaton that escapes parasitism. You will be able to observe how this model may improve upon the Nicholson Bailey model. 3

4 1. Save a copy of your original spreadsheet as Negbinomial. 2. We first add the new parameter k to the Home worksheet in the spreadsheet: 8 binomial power k = E8/10 Slider from 0 to 100, incremental change 1, linked to E8 3. In the Data Worksheet, you must update the cells from rows 6 to 205 with the new formula. 4. After the data values have been updated with the new formula, experiment with different parameter values to begin to observethe behavior of this model. Modify the axes on your three graphs as necessary to obtain meaningful figures. 5. Determine the equilibria for the system and update cells L4 and L5 accordingly. 6. Compute the Jacobian matrix J = J(N,P) and evaluate J at the nontrivial equilibrium ( ˆN, ˆP ). Update the cells B12, C12, B13, C13 accordingly. 7. Using the new spreadsheet investigate the model for each of the parameter value combinations provided above for the Nicholson Bailey model. Take k =0.7 andthenk =0.5. Describe the behavior in each case and compare to the Nicholson Bailey Model. Print each graph. Make a prediction about the stability behavior of the nontrivial equilbrium in general. Can you prove your assertion? A Density Dependent Variation We will now look at another variation of the Nicholson-Bailey model in which we assume that the host population grows to some limiting density. The model becomes: N t+1 = N t exp(r(1 N t /K) ap t ) (1) P t+1 = N t (1 e apt ), (2) where r>0 is a growth constant and K is the carrying capacity. The quantity q = ˆN/K is indicates to what extent the equilibrium population ˆN is depressed by the presence of parasitoids. 1. Save another copy of the Nicholson Bailey file as HostParDep. 2. Update the parameter sliders as follows: 3 Initial Number of Hosts N0 Slider from 0 to 20, incremental change 1, linked to D3 4 Initial Number of Parasitoids P0 Slider from 0 to 20, incremental change 1, linked to D4 5 Host Escape Rate a =E5/100 Slider from 0 to 100, incremental change 1, linked to E5 6 Host Growth Parameter b =E6/10 Slider from 0 to 100, incremental change 1, linked to E6 7 Parasitoid Growth Parameter c =E7/10 Slider from 0 to 100, incremental change 1, linked to E7 8 Growth Parameter r =E8/10 Slider from 0 to 100, incremental change 1, linked to E8 9 Population Depression Factor q =E9/10 Slider from 0 to 100, incremental change 1, linked to E9 Note: we keep b = c = 1 throughout this spreadsheet. Simply leave existing sliders for these parameters set to 1. The nontrivial equilibrium can be expressed as: ˆP = r (1 q) (3) a Update cells L4 and L5 accordingly. ˆN = ˆP 1 e a ˆP (4) 4

5 3. The carrying capacity, K, will be calculuated using the formula K = q ˆN. Input this quantity into cell L3 as: =$L$4/$D$9 Label this cell by entering an appropriate title in cell H3. 4. In the Data Worksheet, you must update the cells from rows 6 to 205 with the new formula. 5. Compute the Jacobian matrix J = J(N,P) and evaluate J at the nontrivial equilibrium ( ˆN, ˆP ). Update the cells B12, C12, B13, C13 accordingly. We will investigate this model in the NP phase plane. Use the new spreadsheet with the following parameter choices. Determine the nontrivial equilibrium ( ˆN, ˆP )andk in each case. Report the moduli of the two eigenvalues of the Jacobian matrix at the equilibrium point. Print each graph of the NP phase plane and the plots of the two populations. Describe the behavior of the dynamics and the stability in each case. Assume that a =0.2 andq =0.4 in each case. (a) r =0.5 (b) r =2 (c) r =2.2 (d) r =2.65 References [1] J.R. Beddington, C.A. Free, and J.H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature 255, 58-60, [2] L. Edelstein-Keshet, Mathematical Models in Biology, Classics in Applied Mathematics 46, SIAM, [3] K.J. Griffiths, The importance of conicidence in the functional and numerical responses of two parasites of the European pine sawfly, Neodiprion sertifer, Canadian Entomologist , [4] R.M. May, Host-parasitoid systems in patchy environments: a phenomenological model, Journal of Animal Ecology , [5] A.J. Nicholson and V.A. Bailey, The balance of animal populations, Proceedings of the Zoological Society of London ,