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Fish Population 1 Introduction this lab, you will use the logistic equation model and study several populations fish living suburban lake. Using basic numerical and qualitative techniques, you will study how the trout population evolve over time both with and without harvesting 2 Mathematical Model Suppose that suburban lake stocked with three types fish: rainbow trout (R). brown trout (T). and bass (B). the absence any predation harvesting each of these populations fish may be modeled with the logistic equation (1) where g(f) the size of the fish population (in hundreds of fish) at any given time (days), the parameter >Ois the initial growth population, and >ois the carrying capacity the population. Keep mind the accuracy any predictions based on the logistic model depends upon whether the parameters and are constant Now suppose that the Lake has been opened for fishing (ie "harvesting") and assume for sim- plicity that humans the only predators the fish. To make sure that the lake over- fished tomodel how fishing It is straightforward to adjust the logistic equation (1) include harvesting. h(f) the amount harvesting taking place at time then one way change the model is Now reasonable assume that amount harvesting depends on the number of nish in the lake. (After all, there are more fish the lake then easier catch them.) So let's measure the amount harvesting terms (amount fish) instead terms off Then the model is (2) A reasonable harvesting function could be pg2 h(g) q+y² (3) where the parameters and represent how good the locals are at catching fish. Finally notice that fwe define the right-hand side of the equation (4) then wecan rewrite the differential equation(2)as s'(t) (g) Section Questions 1. What are the units of the parameters and L? (Remember what the units of and t are.) 2. Use separation of variables solve(1) analytically (i.e. no harvesting) 3. Solve (1)or the general form the (de not use specific values of L). 4. Classify the dific erontial equation(2). Isi linear? Isit homogeneous] What sits degree? i t what does fnon- autonomy mean physically forthis problem)? 5. Set but DO NOT solve the integnals that one gets when sparation variables done on model (2) (i.e., with harvesting). Suggest an analytical technique that could be used to evaluate the integrals. 6. Explore the harvesting function h(g). What happens h(g) as gets very large? What if y close to 07 Does this make sense physically? (It may help totry graphing h(y).) 3 Numerical Investigation of (1) Let p= 1.2 and Since there are three species of fish in the lake we would need to investigate population growth species individually, Suppose fish species have the same natural growth rate 0.65 but the carrying capacities different each the species. Rainbow trout have LR =5.4 brown trout have La 8.1. and have LB = 16.3. Pick one species fish to study i this section Suppose that the lake initialh stocked with 200 your fish species choice. . the interval 10 25] with ster sizes -0,0 to numericall- the and true solution (from 2.2 same graph . Plot the Toplo the direction field you may use any program like. provide method. the file dirfield. Instructions for using dirfield are included separately. Section Questions 1. the result Equestion 2.3 for your chosen fish population classify their stability. 2. Compare the "exact" and your fish population How did step size influence the accuracy of numerical solution? 3. Using your direction field. describe the behavior of solutions for various initial conditions 4 Numerical Investigation of (2) It is difficult (although possible) find the exact solution to equation (2) Furthermore the solution complicated hard understand Instead using the exact solution to (2) we will explore he behavior qualitatively As Section let p : 1.2 and g=l. Suppose all the fish species have the same natural growth rate 0.65 but the capacities are differen each the specles. As before. rainbow trout have LR 5.4. brown trout have La 8.1. and bass have Ln 16.3 . Explore the behavior the logistic model(2) for cad specles finsh (Le. foreach value o L). Plot Fasa function and thisplot equilibrium solutions ((2) (Be sure touse suitable scale when plottings that you don't miss any equilibrium values.) . Notice that some values have different number equilibria than others trialand error find values where fequilibria changes (You'l haveto try values between the given values LT and LR-) There aretwo values thisrange when the number dequilibria change. Each these special values calleda hifureation value, . Now, plot direction folds Fibe different values 5,4 8.1 and 16.3, Toplot the like. Weprovide the file dirfield eparately (Note that you may (very)lange time intervalto Esolutions, Section Questions 1. (a) Describe words what an equilibrium (b) Give a mathematical definition Explain vou can this find the equilibria from the plot /(y). 2. Interpret the function plots and direction fields: (a) What happening population when f(a) when f(y)=07 (b) Use the information from part stable unstable (c) Make sure that your function plots and are consistent with each other Discuss what the equilibria look like your fields Hom can you determine stability from your direction fields? (d) Using your direction fields, what are term behaviors of solutions for various initial How does depend on the initial conditions (pop- ulation time 0) and the value L7 3. Report the two bifurcation values that you found above. Describe how the number equilibria changes each value. 4. Based your results. which species of trout best the lake with? There may be various issues consider explain your decision. 5. Discuss these questions briefly with your own Are there any weaknesses in the model used? How do think could improved? Do you think there are additional effects the model should account for?

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