CS 170: Computing for the Sciences and MathematicsAdministriviaConstrained GrowthUnconstrained GrowthRate of change of populationRate of change of populationDeathBehaviorSlide 9Death, part 2Behavior, part 2If population is greater than M…Continuous logistic equationsDiscrete logistic equationsIf initial population < M, S-shaped graphIf initial population > MEquilibriumStabilitySlide 19HOMEWORK!Constrained GrowthCS 170:Computing for the Sciences and MathematicsAdministriviaLast timeUnconstrained GrowthTodayUnconstrained GrowthHW3 assignedThursday’s class will be in P115Constrained GrowthPopulation growth usually has constraintsLimits include:Food availableShelter/”Room”DiseaseThese all can be encapsulated in the concept of “Carrying Capacity” (M)The population an environment is capable of supportingUnconstrained GrowthRate of change of population is directly proportional to number of individuals in the population (P)where r is the growth rate.rPdtdPRate of change of population D = number of deathsB = number of birthsrate of change of P = (rate of change of B) – (rate of change of D)dPdtdBdtdDdtRate of change of populationRate of change of B proportional to PdBdt rPdPdt rP dDdtDeathIf population is much less than carrying capacity, what should the behavior look like?No limiting pressure!BehaviorIf population is much less than carrying capacity, almost unconstrained modelRate of change of D (dD/dt)0dDdt 0dPdtdBdtDeathIf population is nearing the carrying capacity, what should the behavior look like?Death, part 2If population is less than but close to carrying capacity, growth is dampened, almost 0Rate of change of D larger, almost rate of change BdDdtdBdt rPBehavior, part 2For dD/dt = f(rP), multiply rP by something so thatdD/dt 0 for P much less than MIn this situation, f 0dD/dt dB/dt = rP for P less than but close to MIn this situation, f 1What is a possible factor f?One possibility is P/MdPdt rP dDdtIf population is greater than M…What is the sign of growth?NegativeHow does the rate of change of D compare to the rate of change of B?GreaterDoes this situation fit the model?Continuous logistic equationsdDdt(rP)PMdPdt(rP) (rP)PM r 1PMPDiscrete logistic equationsD  D t  D(t  t)  rP t  t  P t  t MtP  births deathsP  rP t  t  t  rP t   t  P t  t MtP  k(1P t  t M)P t  t , where k  rtIf initial population < M, S-shaped graphIf initial population > MEquilibriumEquilibrium solution to differential equationSolution where derivative is always 0 M is an equilibrium point for this modelPopulation remains steady at that valueDerivative = 0Population size tends to M, regardless of non-zero value of populationFor small displacement from M, P  MStabilitySolution q is stable if there is interval (a, b) containing q, such that if initial population P(0) is in that interval thenP(t) is finite for all t > 0P  qP = M is stable equilibriumThere is an unstable equilibrium point as well…P = 0 is unstable equilibriumViolates P  qHOMEWORK!READ Module 3.3 in the textbookHomework 3Vensim Tutorial #2Due NEXT MondayThursday class in P115 (Lab)Chance to work on HW #3 and ask