COMPUTER EXERCISE: POPULATION DYNAMICS IN SPACE September 3, 2013

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COMPUTER EXERCISE: POPULATION DYNAMICS IN SPACE September 3, 2013 Objectives: Introduction to coupled maps lattice as a basis for spatial modeling Solve a spatial Ricker model to investigate how wave

Cellular automata and coupled map lattices are two approaches for studying population dynamics distributed in space, represented as a grid of adjacent sites.

1 COMPUTER EXERCISE: POPULATION DYNAMICS IN SPACE September 3, 2013 Objectives: Introduction to coupled maps lattice as a basis for spatial modeling Solve a spatial Ricker model to investigate how wave speed changes as a function of dispersal rate and recruitment Introduction Preliminary. Cellular automata and coupled map lattices are two approaches for studying population dynamics distributed in space, represented as a grid of adjacent sites. In either case the study involves evaluating a discrete time dynamical system for the growth of the population at each site followed by a dispersal step wherein individuals are redistributed to nearby sites according to some rule. The difference between cellular automata and coupled map lattices is that in cellular automata the state of the sites is binary (occupied/unoccupied) whereas in a coupled map lattice the population size may be any value. Cellular automata and coupled map lattices differ from the reaction-diffusion model in that space is represented as a collection of discrete sites. These models also differ from most metapopulation models in that sites are considered to be adjacent and dispersal is only to sites within a neighborhood of origination. Cellular automata and coupled map lattices provide a good technique for studying population growth within a continuous space where analytical solutions are intractable (e.g., nonlinearity of equations). Boundary conditions. One issue that always comes up in numerical analysis of spatial population dynamics is what to do at the boundaries of the rectangular grid. One possibility is to allow the left boundary to be adjacent to the right boundary and for the top boundary to be adjacent to the bottom boundary. This assumption allows neatly for a continuous space and implies that the spatial geometry of the system is actually a torus, as if the population dynamics were of a microbial population on the surface of a donut. Illustration of a torus. Image:

2 An alternative we will use here is to assume that the grid represents a habitat patch in a matrix of inhospitable territory, in which case individuals that disperse off the grid never return. Population dynamics. In the assigned reading we looked at the spread of a population that was (locally) growing exponentially and dispersing according to the diffusion assumption, giving rise to the reaction diffusion equation. This scenario yielded traveling waves with an asymptotically constant speed proportional to the square root of the product of the intrinsic rate of increase and diffusivity. c*=2 rd In contrast, it is also known that growth equations with leptokurtic (fat-tailed) dispersal kernels we representing long distance dispersal give rise to accelerating traveling waves. But what happens when population dynamics are density-dependent, for instance in fisheries where population growth is often represented by the Ricker stock-recruitment model? n n+ 1 =n t e r (1 n t/ k ) To answer this question (and introduce along the way such techniques as image plots), we will study a coupled map lattice in which the Ricker equation gives the local population dynamics in each each ( larval recruitment ) followed by a redistribution step ( local dispersal ). Exercise 1. The Ricker model defined above will be central to this lab exercise. Since we will often need to update population size, the first step is to write a function called ricker to update population size according to this equation. Spend a little time familiarizing yourself with this model. Perhaps you will want to simulate some trajectories with different values of the parameter r (which is conventionally assumed to represent potential population growth) and k (which governs density dependence, although not in exactly the same way as the parameter with the same name in the logistic model.) Assume carrying capacity is 100. Plot the so-called stock recruitment curve, the line that relates population size at time t to population size at time t+1 for a range of values of r. (Hint: it may be useful to look at values of r along a logarithmic scale. I used values between 2-2 and 2 1/2 ). What are the equilibrium population sizes of the Ricker model? What values of r give rise to population growth? 2. To confirm that your function is working, plot some trajectories for local population growth. As above, assume carrying capacity is 100; initially set r=0.25. How would you describe the dynamics of this system? It is well known that the Ricker model may produce complex dynamics due to overcompensation. (What do you suppose overcompensation is?) Particularly, chaos is approached via the period doubling bifurcation. Find a value of r at which the Ricker model produces a stable two-cycle. What value do you come up with? 3. Now we are in a position to proceed with our coupled map lattice. To start we need to set up an array that we will think of as the space on which population dynamics are occurring. So that your computations don't take too long, I would recommend a lattice no larger than 100,000 sites. (I used 1,000 sites as a trial and that worked just fine.) Use the function matrix to set up a square array of the right dimensions. For later use, initialize the array with the value 0 at all

3 sites. 4. Now, unless you did something unusual in writing the function ricker, it should actually be able to update the entire matrix in one step. That is, if you have an array N that contains the population size at each site, you should be able to compute the population size at the next time step using the code N< ricker(n,r=0.25,k=100). What we lack, however, is a function to update the spatial distribution after dispersal. To be more concrete, our target a computer program executing the following operations: Initialize variables For all times iterate the following i. update population size ii. redistribute iii. store output Inspect results Therefore, the next step is to write a function dispersal that will accomplish the redistribution in step (ii). Assume that this function will take two arguments, the current state of the system (your array N) and a site-to-neighbor dispersal rate m. (We will assume that dispersal is to the von Neumann neighborhood, the four cells orthogonally surrounding a central cell.) A word to the wise: this is probably the trickiest part of the whole exercise. (Hint: Pay close attention to the boundaries. Double hint: let the outermost border of sites be hostile matrix. This simplifies the code considerably). 5. Now you have all the conceptual tools you need to study the Ricker coupled map lattice and do determine its spread rate. Here are a couple of programming tools that you may find helpful. The function image takes a matrix and makes a heat map from it. This is useful for visualizing the state of the system. The operator %in% allows you to test if a value is an element of a set. My code uses the following line to plot the state of the system only every so often rather than at each update step: if(t %in% plot.times) image(n, col=heat.colors(100), zlim=c(0,100)) The function par allows one to reset graphical parameters. Particularly, the argument mfrow lets one set up a multi-panel plot by specifying a grid (composed of a number of rows and a number of columns) of individual plots that are successively populated. My code uses the following line to achieve this. x11(w=12, h=12); par(mfrow=c(5,5)) Starting with the values used above (r=0.25, k=100) and assuming that 1% of individuals move in each of the cardinal directions, iterate the coupled map lattice and watch the invasion play out (this will take a couple hundred time steps). If your analysis generates that following plot, then you have been successful!

4 6. Now you are prepared to answer our question: How does invasion speed change over time in the Ricker coupled map lattice? Answering this question will require (1) developing a method to track the maximum distance the population has spread from the origin, and (2) inspecting how this quantity changes over time. (Hint: A construction nesting two R functions may be helpful here, i.e., min(which(...)) where the ellipsis indicates that I have left out some of the code.) How does spread rate change over time? Does the invasion reach a constant speed? Accelerate? Decelerate? How does spread rate change with different levels of population growth (different values of r)? Different levels of dispersal (different values of m)?

5 Different levels of density dependent (different values of k)? 7. Another useful visualization is the profile of population size along a transect, i.e., if I were to choose a starting point and walk across the space in one direction (a transect) measuring the local abundance of the population as I go, how would abundance change in the course of my trip? Return to your main program and seek do develop such a profile plot. Compare this result with the image plot obtained above. Do these show different pieces of information. 8. If you've made it this far you've come a long way toward understanding how invasion speed is affected by density-dependent population growth. The Ricker model allows us to ask one more question however. As we discovered above and will study in a more thorough way in a few weeks, the Ricker model allows for persistent oscillations. This raises the question about how fast invasions will happen when dynamics are complex as a result of overcompensation? To answer this question, iterate the coupled map lattice using the value of r for which you previously showed there to be a stable two-cycle. Generate the corresponding image plots and profile plots. What is the effect of overcompensation on the invasion process? To demonstrate your work, turn in written answers to these questions and any plot that may help illustrate how you arrived at your answer. Due: September 10, 2013

CELLULAR AUTOMATA IN MATHEMATICAL MODELING JOSH KANTOR. 1. History

CELLULAR AUTOMATA IN MATHEMATICAL MODELING JOSH KANTOR 1. History Cellular automata were initially conceived of in 1948 by John von Neumann who was searching for ways of modeling evolution. He was trying