Introduction to Population Dynamics

Transcription

1 MATH235: Differential Equations with Honors Introduction to Population Dynamics Quote of Population Dynamics Homework Wish I knew what you were looking for. Might have known what you would find. Under the Milky Way Tonight : The Church (1988) 1. Introduction The following problems treat the modeling of dynamics populations. The modeling perspective begins with a set of assumptions, which yield simple models and from these simple models derive more complicated ones. Eventually, we study some fairly recent mathematical models associated with tumor growth. It is important to notice that as we try to model more complicated phenomenon the models naturally become nonlinear. Also, as the equations become nonlinear they become more difficult to mathematically analyze. This is generally the case. Lastly, though we focus our attention on population models, it is important to remember that the methods used and described here are common tools used to study first-order ODEs. I try to keep this in mind while discussing facets of the problems. 2. Linear Growth Models It is always natural to start off in the linear regime since we have a standard set of tools to analyze linear problems. 1 We did something like this in class but now let s be a little more general. Again we will have a general three-part process: (1) Create the mathematical model from a list of empirical assumptions. (2) Analyze the mathematical model through quantitative, qualitative and approximation techniques. (3) Interpret the predictions of the model and refine the assumptions and model as needed. With this in mind we begin the modeling process Modeling - Part I: Assumptions and Mathematical Model. Suppose we have a population, P, that is a function of time, t, and is assumed to grow at a rate proportional to the current population. Also, assume that the growth rate, k, and harvesting rate, H, are functions of time Model Equation. Write down a differential equation that models this population dynamic Modeling - Part II: Mathematical Statement and Analysis. Hopefully you arrived at a standard first-order linear ODE. The major difference between your model and the one from class is that the proportionality constant depends on time and also that the system is subject to some amount of harvesting loss. We now analyze this equation General Solution. Report the general solution to this problem in terms of integrals Constant Growth Rate with Constant Harvesting. Suppose that the growth rate is given by k(t) = k and that the harvesting rate is given by H(t) = H where k, H R +. Find a relationship between k and H such that the population remains positive for all time. 1 It is important to note that our standard set of tools seems bullet-proof right now but this won t last. Higher order problems, even linear ones, still have open questions but that will have to wait. For now just be happy that integrating factors always exist. 2 It s important to note that we are not trying to use a phase line at this point. Remember that a phase line is created by knowing equilibrium solutions and equilibrium solutions are constant solutions to the differential equation. If a problem is not autonomous then there cannot be constant solutions, which satisfy the ODE and thus no phase line arguments could occur. 1

2 Periodic Growth Rate with Constant Harvesting. Suppose that the growth rate is given by k(t) = cos(t) and the harvesting rate is constant H(t) = 1. Using graphing aids, analyze the dynamics of initial populations between and 1.5. Sketch or print out the associated graphs and comment on what you see Periodic Growth Rate with Periodic Harvesting. Suppose that the growth rate is given by k(t) = cos(t) and the harvesting rate is given as H(t) = cos(t). Using graphing aids, analyze the dynamics of initial populations between 0.5 and Sketch or print out the associated graphs and comment on what you see Modeling - Part III: Interpretation, Conclusions and Refinement. Though we introduced harvesting and non-constant growth rates the basic nature of the base autonomous model did not change much Comments. Based on what you saw from the previous analysis, what do you think the linear model does a good job of? What do you think it doesn t do a good job of? How would you change the model to strengthen the models weaknesses? 3. Logistics Growth Models Based on the last section we can conclude that the linear model does not capture any sort of global bound on the population. Empirically, we know that such a bound exists and with this in mind we would like to make a model predicts one. The mentality is to start with the previous model and tweak it into a form that predicts what we see. In class we adopted the logistics growth model, ( dp dt = kp 1 P ) (1), k, N R +. N We studied this equation in class through phase lines and separation of variables. Now we are going to make some natural generalizations Generalization of Logistics Growth. It makes sense to think that the growth rate k and carrying capacity N are time dependent. 3 Also, it makes sense to consider a population of this type being harvested for some reason and that this harvesting is also time dependent Vocabulary. Alter (1) to take into account these new refinements and after this apply our current vocabulary words to describe this equation Analytic Tools. What solution techniques, if any, do we have that can be applied to this equation? 4 When possible solve this equation Autonomy. The previous problem shows how difficult it can be to analytically solve a general nonautonomous nonlinear ODE. Just a simple harvesting term breaks the only method available analytic tool. To get around this we return to the autonomous case Phase Line Analysis. Analyze the logistics problem for constant growth rate, carrying capacity and harvesting rate. Specifically, assume that the logistics model is autonomous and conduct a phase line analysis of the differential equation. Also, show that the solution curves between zero and the carrying capacity have an inflection point at N/ Bifurcation. For what harvesting value does the differential equation undergo a bifurcation? What type of bifurcation occurs? 3 It also makes sense to think that they are not! Consider a petri dish stained with a culture that is left outside. The growth of the cells depends not only on the population but also the ambient temperature of the environment. If left outside, we can expect a time dependence in this temperature and consequently the growth rate. However, we don t leave a petri dish outside and instead keep it in some sort of incubator, which keeps the temperature/growth rate constant. The moral here is that one can apply an external condition so that the material parameters are time independent. Another way of thinking about this is to keep the petri dish outside for only a short amount of time. During this time the temperature should be mostly constant and thus time independent. That is, there exists some time scale for which the material parameters are time independent. In other words, there is not a lot of harm in studying autonomous problems as opposed to nonautonomous ones. 4 There is some amount of hope in the methods for Bernoulli equations. Not a lot, but some.

3 Harvesting Problems and Graphical Help. If the population falls to near zero because the harvesting level H is slightly greater than kn/4, then why must harvesting be banned completely in order for the population to recover? That is, if a level of harvesting just above H = kn/4 causes a near-collapse of the population, then why can t the population be restored by reducing the harvesting level to just below H = kn/4? Separation of Variables: Partial Fractions. Solve the autonomous logistics growth problem via separation of variables and partial fraction decompositions Separation of Variables: Hyperbolic Functions. Complete the square and solve the previous problem via hyperbolic functions. 4. Logistics Growth: Modified Harvesting We see in section that, if unchecked, constant harvesting can lead to the destruction to a population. This problem seeks to find a better way. The idea starts with a simple assumption New Assumption. Assume that the harvesting is proportional to the current population with rate constant α R +. Adjust the previous logistics growth model to harvest in this way as opposed to a constant or time-dependent way Phase Line Analysis. Conduct a phase line analysis for this logistics equation with modified harvesting Bifurcation. For what value of α does a bifurcation occur? Harvesting Dynamics. Suppose we increase α in small steps, allowing the population to reach equilibrium after each step. How do the population dynamics change as α increases? Harvesting Differences. Under the same assumptions of 3.2.3, describe the differences between the long term population dynamics of the species modeled by the previous harvesting model and the species modeled by here in terms of each systems bifurcation value? Which model is more sensitive to over-harvesting? (2) 5. Logistics Growth: Critical Mass Assume that a population P obeys the mathematical model, ( dp = fn (P ) = kp 1 P ) ( ) P dt N M 1 where k, N, M R + such that 0 < M N Interpretation and Analysis. This model has been used to describe populations of squirles in the Rocky Mountains. The modification of logistics growth makes this differential equation increasingly nonlinear but since it is still autonomous the analysis is similar to the previous problems Modeling. Describe the possible physical meaning of the parameter M and the affect the term ( P M 1) has on the population dynamics Fixed Points and Bifurcation. Sketch the graph of f N (P ) for various values of N. At what value of N does a bifurcation occur? Qualitative Analysis. How does the population dynamics change if the parameter N is slowly and continuously decreased through the bifurcation value? Graphical Analysis. Set k = 1, M = 2, and using a graphing package, make three different graphs of the slope field for (2) for N 1 = 1, N 2 = 2, N 3 = 3. On each of these graphs plot the unique solution, which satisfies the initial condition, P i = (t 0, y 0), i = 1, 2, 3, 4, 5, 6, where the P i s are given by P 1 = (0,.5), P 2 = (0, 1), P 3 = (0, 1.5), P 4 = (0, 2), P 5 = (0, 2.5), P 6 = (0, 3). Comment on the results. 5 To see this it is useful to graph solutions to the ODE for various values of the parameters. See the blog for help with graphical tools.

4 4 6. First Order Autonomous Polynomial Equations The previous problems illustrate the refinements gained by increasing the nonlinearity of the autonomous ODE. Let s take this to a natural conclusion by considering the autonomous ODE (3) where (4) f(y) = dy dt = f(y) N a iy i, N N +. i= Fundamental Assumption. Assume that the previous polynomial as N many real, distinct and positive roots, 6 r i, that can be well ordered as (5) r 1 < r 2 < r 3 < < r N Separation of Variables. Using this information, separate variables and prepare the associated integrals Integrations. Using partial fractions, conduct the integrations found in previous step Refinements. Lastly, show that the implicit solution to the ODE takes the form (6) where (7) N (r i y) ( 1)i+1 γ i = Ce t, C R i=1 γ i = N (r j r i) 1 j=1 j i 7. Tumor Growth The Gompertz-Makeham law states that the death rate is the sum of an age-independent component (Makeham term) and an age-dependent component (the Gompertz function), which increases exponentially with age. 7 The latter was proposed by Benjamin Gompertz in 1825 and has since become frequently used to describe tumor growth. This law can be written in differential form as 8 (8) dn dt = N [a b ln(n)], where a, b R + and N represents the number of individuals in the population Model Adaptation. Given (8), introduce a loss(harvesting) term associated with the main equation from Mathematical Modeling and Cancer, SIAM News, Volume 37, Number 1, Model Equivalence. Determine the relationship between the constants a and b from (8) and θ and r from the SIAM article so that the model equations are equivalent Model Interpretation. Explain the physical meaning of the constants θ and r from the SIAM model. Also, what does the variable N represent? Loss Function. It is stated in the SIAM article that Fister and Panetta considered three possible scenarios for G(N, u(t)). Provide possible functional forms for G for each of the three scenarios. 6 This is not needed but will make the calculation easier. One can think of this assumption as the natural extension of the previous refinements to the logistics model. First, the dependent variable is assumed to be positive and second, the increase of roots corresponds to changes dynamics similar to the lower bound for growth, M, seen before. 7 Taken from wikipedia 8 This equation is taken from A generalization of Gompertz law compatible with the Gyllenberg-Webb theory for tumour growth., d Onofrio, Fasano and Monechi, Mathematical Biosciences However, a simplification of this appears in Strogatz and this equation is roughly the same as what is found in the posted SIAM article, up to some log manipulations. 9 Actually, N can represent any quantity associated with the size of the population. Maybe the associated volume for example.

5 Model Analysis. Mathematically, it is somewhat easier to conduct analysis on (8) and like before we simplify the model to its autonomous form. 10 Specifically, we consider (8), as written, and without any loss/treatment component Phase Line Analysis. Find all equilibria and, using a phase line, graph possible solutions to (8) Separation of Variables. Solve (8) Graphical Analysis. Consider logistics growth without harvesting and where r = 1, N = e and (8) where a = b = 1 and graph solutions to the two equations using the same initial values. Comment on the results Treatment Analysis. In the previous section you considered some options for G(N, u(t)). It is likely that your choices caused the ODE to be either not separable or not integrable. While this is a problem for our hands it is not a problem for our numerical methods Graphical Help: Part I. Pick one of your choices and, using a numerical approximation, graph the associated solution. What was the outcome? Is it what you expected? Graphical Help: Part II. Based on Part I, refine your choice for G. Describe your thought process and its graphical outcome. What would you do next? 10 It is interesting to note that this form is similar to the logistics equation where the dependent variable, associated with carrying capacity, is replaced by the log of that variable.