1 Chapter 5 Biology 5 1 Mathematical models are used to predict the growth of a population that is population size at some future date The simplest model is that for exponential growth The calculation requires knowledge of the organism s maximum specific growth rate A value for this coefficient can be obtained from field observations of population size or from laboratory experiments where population size is monitored as a function of time when growing at high substrate concentrations i e S Ks Time d Biomass mg L 0 50 1 136 2 369 3 1 004 4 2 730 5 7 421 Calculate max for this population assuming exponential growth include appropriate units Solution The exponential growth model can be log linearized In this case using all of the data a plot of ln Xt versus t would yield max as the slope ln X t max t ln X This would provide the most accurate value of max Alternatively a value for max may be calculated using the analytical solution to the exponential growth model X t X 0 e max t Rearrange and solve for max max ln X t X 0 t Perform the calculation using any two measures of biomass and the associated time max mg mg 50 ln 7 421 L L 1 0 day 5 days 0 days Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 2 5 2 Once a value for max has been obtained the model may be used to project population size at a future time Assuming that exponential growth is sustained what will the population size in Problem 5 1 be after 10 days Solution Again using the exponential growth model and the values for max determined in Problem 5 1 with Xo 50 mg L t 10 days and max 1 day X 10 50 mg 1 day 10 days mg e 1 101 323 L L Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 3 5 3 Exponential growth cannot be sustained forever because of constraints placed on the organism by its environment that is the system s carrying capacity This phenomenon is described using the logistic growth model a Calculate the size of the population in Problem 5 1 after 10 days assuming that logistic growth is followed and that the carrying capacity is 100 000 mg L b What percentage of the exponentially growing population size would this be Solution a The population size after 10 days according to the logistic growth model can be calculated using the analytical solution with K 100 000 mg L and Xo and max as given in Problem 5 1 Xt K K X o max t 1 e Xo mg mg L 91 680 Xt mg mg L 100 000 L 50 L 1 day 10 day 1 e mg 50 L 100 000 b Problem 5 2 provided the estimated population at 10 days This value needs compared to the carrying capacity value determined in part a of this problem 91 680 100 8 1 101 323 Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 4 5 4 Food limitation of population growth is described using the Monod model Population growth is characterized by the maximum specific growth rate max and the half saturation constant for growth Ks a Calculate the specific growth rate of the population in Problem 5 1 growing at a substrate concentration of 25 mg L according to Monod kinetics if it has a Ks of 50 mg L b What percentage of the maximum growth rate for the exponentially growing population size would this be Solution a The specific growth rate for a population growing under nutrient limited conditions can be calculated using the Monod model max S Ks S 1 day 25 mg L mg mg 50 25 L L 0 33 day b Problem 5 1 provided the estimated maximum growth rate at 1 day This value needs compared to the specific growth rate determined in part a of this problem 0 33 day 100 33 1 day Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 5 5 5 Laboratory studies have shown that microorganisms produce 10 mg L of biomass in reducing the concentration of a pollutant by 50 mg L Calculate the yield coefficient specifying the units of expression Solution Remember that by definition Y dX dt 10 mg biomass L mg biomass 0 2 dS dt 50 mg substrate L mg substrate Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 6 5 6 When food supplies have been exhausted populations die away This exponential decay is described by a simple modification of the exponential growth model Engineers use this model to calculate the length of time that a swimming beach must remain closed following pollution with fecal material For a population of bacteria with an initial biomass of 100 mg L and a kd 0 4 day calculate the time necessary to reduce the population size to 10 mg L Solution Under conditions of exhausted food supplies S 0 the overall population growth model is used dX X S max 1 kd X dt K Ks S However because S equals zero the above expression reduces to a first order decay dX kd X dT The above expression can be solved X t X o e kd t And this expression can be solved for time t Then given values can be substituted as follows Xo 100 mg L Xt 10 mg L and kd 0 4 day X 10 mg L ln t ln 100 mg L Xo t 5 8 days 0 4 day kd Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 7 5 7 A population having a biomass of 2 mg L at t 0 days reaches a biomass of 139 at t 10 days Assuming exponential growth calculate the value of the specific growth coefficient Solution X t X o e t ln Xt t Xo ln 139 mg L 10 days 2 mg L 0 42 day Solutions Manual prepared by Ziad Katirji and Heather E Wright Wendel Environmental Engineering Fundamentals Sustainability Design James R Mihelcic and Julie Beth Zimmerman John Wiley Sons New York 2009 8 5 8 Fecal bacteria occupy the guts of warm blooded animals and do not grow in the natural environment Their population dynamics in lakes and rivers that is following a discharge of raw sewage can be …