Modeling Populations: an introduction AiS Challenge Summer Teacher Institute 2003 Richard Allen Population Dynamics Studies how populations change over time Involves knowledge about birth and death rates, food supplies, social behaviors, genetics, interaction of species with their environments and among themselves. Models should reflect biological reality, yet be simple enough that insight may be gained into the population being studied. Overview Illustrate the development of some basic one- and two-species population models. Malthusian (exponential) growth – human populations Logistics growth – human populations Logistics growth with harvesting. Predator-Prey interaction – two fish populations The Malthus Model In 1798, the English political economist, Thomas Malthus, proposed a model for human populations. His model was based on the observation that the time required for human populations to double was essentially constant (about 25 years at the time), regardless of the initial population size. US Population: 1650-1800 U.S. population data available to Malthus. Governing Principle To develop a mathematical model, we formulate Malthus’ observation as the “governing principle” for our model: Populations appeared to increase by a fixed proportion over a given period of time, and that, in the absence of constraints, this proportion is not affected by the size of the population. Discrete-in-time Model …, tN: equally-spaced times at which the t0, t1, t2, population is determined: dt = ti+1 - ti P0, P1, P2, …, PN: corresponding populations at times t0, t1, t2, …, tN b and d: birth and death rates; r = b – d, is the effective growth rate. P0 P1 P2 … PN |---------|---------|----------------|-----> t t0 t1 t2 … tN The Malthus Model Mathematical Equation: (Pi + 1 - Pi) / Pi = r * dt r=b-d or Pi + 1 = Pi + r * dt * Pi ti+1 = ti + dt The initial population, P0, is given at the initial time, t0. An Example Example: Let t0 = 1900, P0 = 76.2 million (US population in 1900) and r = 0.013 (1.3% per-capita growth rate per year). Determine the population at the end of 1, 2, and 3 years, assuming the time step dt = 1 year. Example Calculation P0 = 76.2; t0 = 1900; dt = 1; r = 0.013 P1 = P0 + r* dt*P0 = 76.2 + 0.013*1*76.2 = 77.3; t1 = t0 + dt = 1900 + 1 = 1901 P2 = P1 + r* dt*P1 = 77.3 + 0.013*1*77.3 = 78.3; t2 = t1 + dt = 1901 + 1 = 1902 P3 = P2 + r* dt*P2 = 78.3 + 0.013*1*78.3 = 79.3; t3 = t2 + dt = 1902 + 1 = 1903 Pi = ?, ti = ?, i = 4, 5, … US Population Prediction: Malthus Malthus model prediction of the US population for the period 1900 - 2050, with initial data taken in 1900: t0 = 1900; P0 = 76,200,000; r = 0.013 Actual US population given at 10-year intervals is also plotted for the period 1900- 2000 Malthus Plot Pseudo Code INPUT: t0 – initial time P0 – initial population dt – length of time interval N – number of time steps r – population growth rate Pseudo Code OUTPUT ti – ith time value Pi – population at ti for i = 0, 1, …, N ALGORITHM: Set ti = t0 Set Pi = P0 Print ti, Pi Pseudo Code for i = 1, 2, …, N Set ti = ti + dt Set Pi = Pi + r*dt*Pi Print ti, Pi end for Logistics Model In 1838, Belgian mathematician Pierre Verhulst modified Malthus’ model to allow growth rate to depend on population: r = [r0 * (1 – P/K)] Pi+1 = Pi + [r0 * (1 - Pi/K)] * dt * Pi r0is maximum possible population growth rate. K is called the population carrying capacity. Logistics Model Pi+1 = Pi + [r0 * (1 - Pi/K)] * dt * Pi ro controls not only population growth rate, but population decline rate (P > K); if reproduction is slow and mortality is fast, the logistic model will not work. K has biological meaning for populations with strong interaction among individuals that control their reproduction: birds have territoriality, plants compete for space and light. US Population Prediction: Logistic Logistic model prediction of the US population for the period 1900 – 2050, with initial data taken in 1900: t0 = 1900; P0 = 76.2M; r0 = 0.017, K = 661.9 Actual US population given at 10-year inter- vals is also plotted for the period 1900-2000. Logistic plot Logistics Growth with Harvesting Harvesting populations, removing members from their environment, is a real-world phenomenon. Assumptions: Per unit time, each member of the population has an equal chance of being harvested. In time period dt, expected number of harvests is f*dt*P where f is a harvesting intensity factor. Logistics Growth with Harvesting The logistic model can easily by modified to include the effect of harvesting: Pi+1 = Pi + r0 * (1 – Pi / K) * Δt * Pi - f * Δt * Pi or Pi+1 = Pi + rh * (1 – Pi / Kh) *Δt * Pi where rh = r0 - f, Kh = [(r0 – f) / r0] * K Harvesting A Predator-Prey Model: two competing fish populations An early predator-prey model In the mid 1920’s the Italian biologist Umberto D’Ancona was studying the results of fishing on population variations of various species of fish that interact with each other. He came across data on the percentage-of-total- catch of several species of fish that were brought to different Mediterrian ports in the years that spanned World War I Two Competing Fish Populations Data for the port of Fiume, Italy for the years 1914 -1923: percentage-of-total-catch of predator fish (sharks, skates, rays, etc), not desirable as food fish. Fium e, Italy 40 selachians Percent 30 20 Fium e, Italy 10 0 1910 1915 1920 1925 Years Two Competing Fish Populations The level of fishing and its effect on the two fish populations was also of concern to the fishing industry, since it would affect the way fishing was done. As any good scientist would do, D’Amcona contacted Vito Volterra, a local mathematician, to formulate a model of the growth of predators and their prey and the effect of fishing on the overall fish population. Strategy for Model Development The model development is divided into three stages: 1. In the absence of predators, prey population follows a logistics model and in the absence of prey, predators die out. Predator and prey do not interact with each other; no fishing allowed. 2. The model is enhanced to allow for predator- prey interaction: predators consume prey 3. Fishing is included in the model Overall Model Assumptions Simplifications Only two groups of fish: prey (food fish) and predators. No competing effects among predators No change in fish populations due to immigration into or emigration out of the physical region occupied by the fish. Model Variables Notation ti - specific instances in time Fi - the prey population at time ti Si - the predator population at time ti rF - the growth rate of the prey in the absence of predators rS - the growth rate of the predators in the absence of prey K - the carrying capacity of prey Stage 1: Basic Model In the absence of predators, the fish population, F, is modeled by Fi+1 = Fi + rF * dt * Fi * (1 - Fi/K) and in the absence of prey, the predator population, S, is modeled by Si+1 = Si –rS * dt * Si Stage 2: Predator-Prey Interaction a is the prey kill rate due to encounters with predators: Fi+1 = Fi + rF*dt *Fi*(1 - Fi/K) – a*dt*Fi*Si b is a parameter that converts prey-predator encounters to predator birth rate: Si+1 = Si - rS*dt*Si + b*dt*Fi*Si Stage 3: Fishing f is the effective fishing rate for both the predator and prey populations: Fi+1 = Fi + rF*dt*Fi*(1 - Fi/K) - a*dt*Fi*Si - f*Δt*Fi Si+1 = Si - rS* dt *Si + b*dt*Fi*Si - f*dt*Si Model Initial Conditions and Parameters Plots for the input values: t0 = 0.0 S0 = 100.0 F0 = 1000.0 dt = 0.02 N = 6000.0 f = 0.005 rS = 0.3 rF = 0.5 a = 0.002 b = 0.0005 K = 4000.0 S0 = 100.0 Predator-Prey Plots D’Ancona’s Question Answered (Model Solution) A decrease in fishing, f, during WWI decreased the percentage of equilibrium prey population, F, and increased the percentage of equilibrium predator population, P. f Prey Predators 0.1 800 (82.1%) 175 (17.9%)+ 0.01 620 (74.9%) 208 (25.1%) 0.001 602 (74.0%) 212 (26.0%) 0.0001 600 (73.8%) 213 (26.2%) + (%) - percentage-of-total catch Reference URLs Shodor site: Predator-Prey models www.shodor.org/scsi/handouts/twosp.html More discussion about the Fiume fish catch http://www.math.duke.edu/education/webfe ats/Word2HTML/Predator.html Google: Search for “population models”, predator-prey models”, etc.