Chapter 2: Growth & Decay

Transcription

1 Chapter 2: Growth & Decay 107/226 Introduction In this chapter, model biological systems whose population is so large or where growth is so fine-grained that continuity can be assumed. These continuous models give rise to differential equations. As a consequence, there is no natural time delay in the models and hence none of the over-compensation of difference equations. Extend the logistic growth equation model above & add a culling term. Develop the ideas from this to a non-linear model for the chemostat. 108/226

2 Simple Models For cell pop n whose env t doesn t alter with time (i.e. nutrients/ space), can assume that rate of increase of cells number of cells. Mathematically, can put this as: dn = rn (3.1) dt for some constant of proportionality, r. r measures average growth rate per unit time per individual (i.e. 2.4 children per person per life number). This equation (a.k.a.the law of organic growth) was named after Malthus 4 a gloomy philosopher called who predicted (ca 1798) that world pop n would soon outgrow its resources. 4 T.R. Malthus ( ) 109/226 Simple Models cont d In real life experiments there is always a finite starting value of N at t = 0, N 0. Can show that: N(t) = N 0 e rt (3.2) i.e. (given space/ food) growth of cells is exponential. Note that the value of the constant r can be -ive or +ive, corresponding to decay or growth respectively. For a human population with α births per year & β deaths, obviously r = α β & population viability depends on this being +ive. Can be shown from Eqn.(3.2) that doubling time for pop n given by τ 2 = ln2/r so (if r is a percent) world s population doubles itself every 70/r years. 110/226

3 Restricted Growth: the Verhulst or logistic model Pop ns do not grow in above manner; restrictions (due to food, space, predation or any other factors) apply on all growth rates. These put a limit on max sustainable pop n size. This limit is called the carrying capacity of environment, K. Instead of constant r given by Eqn.(3.1), have a time dependent growth rate 1 dn N dt. Eqn.(3.1) implies a constant growth rate given by r. 111/226 Restricted Growth cont d In logistic growth model, growth rate is assumed to start at r when N = 0 & reach zero when N = K, ( Growth Rate 1 N ) K i.e. the growth rate decreases linearly with increasing population. Mathematically logistic growth model is given by: 1 dn N dt = r ( 1 N K This differential equation involves parameters r, K: r is relevant to initial growth phase of a pop n (before restrictions have an effect) K imposes an upper limit on population growth. ) (3.3) 112/226

4 Applications of the Model: r/k selection theory In ecology, so-called r/k selection theory, relates to selection of combinations of traits in a species that trade off btw quantity or quality of offspring. r,k refer to low- & high-density conditions, respectively. Terminology came about from the work of the ecologists MacArthur & Wilson based on their work on insulated ecosystems. Species are often called either r-strategist or K-strategist depending upon the selective processes shaping their reproductive strategies. Theory contends that adaptation to high or low density environments involve different characteristics. 113/226 Applications of the Model: r/k selection theory cont d r-selection In unstable/unpredictable environments, r-selection is dominant mechanism because ability to reproduce quickly is crucial. Adaptations that permit successful competition with other organisms are unnecessary, due to environmental volatility. Characteristic traits of r-selection are thought to include: short generation time, high fecundity & ability to disperse offspring widely, early maturity onset with small body size. Examples range from bacteria, through insects (e.g. mosquitos) & weeds, to various mammals, especially small rodents (e.g. rats). 114/226

5 Applications of the Model: r/k selection theory cont d K-selection In stable/predictable environments, K-selection is dominant mechanism because ability to compete successfully for limited resources is crucial. Pop n sizes of K-selected organisms typically very stable & approach max that the environment can bear. Differ from r-selected populations, where population numbers are more volatile. Characteristic traits of K-selection are thought to include: long life expectancy, mate choice, production of fewer offspring requiring extensive parental care until maturity & large body size. Examples large organisms such as elephants, trees, humans & whales, but also some smaller organisms. 115/226 Further Applications of the Model As well as above, Verhulst Model has been applied to other areas: 1 Spread of a Disease. Initial no. of susceptible individuals to an infectious disease in pop n is (a const) K. So y & (K y) are no. of infectives & susceptibles after t. Chance encounters spread disease at a rate proportional to (product of) infectives & susceptibles. So model is ẏ = ry(k y) Spread of rumors is an example of an identical model to this. 116/226

6 Further Applications of the Model cont d 2 Explosion/Extinction. No. y(t) of crocs in a swamp satisfies ẏ = ry where growth-decay constant r (y M) & M is a threshold population. The logistic model ẏ = k(y M)y gives extinction for initial populations smaller than M & a doomsday population explosion y(t) for y(0) > M. This model ignores harvesting/culling. 117/226 Further Applications of the Model cont d 3 Logistic Growth with Culling. No. y(t) of fish in a lake satisfies a logistic model ẏ = (a by)y c, provided fish are harvested at a rate c > 0. This rate c can be either constant, variable (i.e. c = hy) or involve periodic harvesting & restocking c = hsin(ωt). These models are dealt with in more detail below 118/226

7 Further Applications of the Model cont d 4 The Chemostat Model. A chemostat (from Chemical environment is static) is a bioreactor taking in fresh medium continuously, while culture liquid is continuously removed to keep the culture volume constant. By changing rate with which medium is added to bioreactor, growth rate of microorganism can be easily controlled. This model is examined below. 119/226 Restricted Growth cont d Solving the general form of logistic growth equation Eqn.(3.3), (left as an exercise) to get: N(t) = K ( K N0 1 ) e rt +1 (3.4) for some inital population N 0. Can be seen from Figs 3.1(a) & (b) that logistic growth of pop n depends qualitatively on the initial population: For N 0 > K/2, second derivative N is always -tive, For N 0 < K/2 there is a period of growth which is exp l in nature ( N > 0) which then levels off ( N = 0) before heading asymptotically to K ( N < 0) as t 120/226

8 Restricted Growth cont d Population Size, N Population Size, N Growth Rate, k Time, t Growth Rate, k Time, t (a) N 0 > K/2 (b) N 0 < K/2 Figure 3.1 : Logistic Growth for Different N 0 121/226 Restricted Growth cont d Taking Eqn.(3.3) & adding a culling/ harvesting term H: ( dn = rn 1 N ) H (3.5) dt K which may be non-dimensionalised (removal of dimensions, by dividing through by carrying capacity, K), as follows: dy = ry (1 y) h (3.6) dt where y = N/K & h = H/K are non-dimensional pop n & culling rates resp. Can get steady states of this eqn (all y for which dy/dt = 0): ( dy = 0 hence r y 2 y + h ) = 0 (3.7) dt r 122/226

9 Restricted Growth cont d Provided they meet the harvesting condition 4h < r, both roots of Eqn 3.7: ( y = p ± = 1 ) 1± 1 4h (3.8) 2 r are steady states since both have the property y = N/K < 1. Note: condition 4h < r gives us population harvesting capacity (i.e. how much may be harvested). In order to find stable states of Eqn.(3.6), it is necessary to graph the derivative of y against y itself. This allows us to examine the behaviour of the derivative for various (especially initial) values of y. 123/226 Restricted Growth cont d Examining the graph (Fig. 3.2), see that points p and p + from Eqn.(3.8) demarcate areas of stability. For y(0) < p, pop n will never grow, as ẏ < 0 for all t and thus extinction will occur. For y(0) > p, ẏ > 0 & pop n will eventually settle down to a steady state y = p + (this follows from working out a solution to Eqn.(3.6) and examining behaviour of y(t) as t ). Can be seen from Fig 3.2, that the time of optimum culling is the point where ẏ is maximal. Before this pop n is not growing at an optimal rate. After this growth rate is decreasing to the steady state. 124/226

10 Restricted Growth cont d dy dt y(t) y = p y = p + Figure 3.2 : ẏ V y for Logistic Growth with Culling 125/226 Aside: Equilibrium & Stability A constant solution of a differential equation is called an equilibrium solution or steady state. Locating/ classifying steady states helps determine family of all solutions of DE equation. Soln to a DE of form N(t) = constant, because dn/dt = 0 is called an equilibrium point. 126/226

11 Aside: Equilibrium & Stability cont d Classify steady states according to behavior of other solutions that start nearby: Steady states N = c is called stable if any solution N(t) that starts near N = c stays near it. Equilibrium N = c is called asymptotically stable if any solution N(t) that starts near N = c actually converges to it, that is lim t = c. If an equilibrium state is not stable, it is called unstable. This means there is at least one solution that starts near equilibrium & runs away from it. Example of this is first state N = 0 of the Logistic Growth model with no culling (similar to Fig. 3.2, only shifted to the left to have roots at N = 0 and N = K) Eqn.(3.3), this is an unstable state. This is because, if N = +ǫ (for small ǫ), the population will increase; if N = ǫ, the population will decrease further. 127/226 The Chemostat Logistic Growth model can be used for Chemostat, (used to study growth of µorganisms under various lab conditions). For the growth models so far, since the nutrients are not renewed, exp l growth is limited to a few generations. Bacterial cultures can be maintained in a state of exp l growth for long time using a system of continuous culture, designed to relieve the conditions that stop exp l growth in batch cultures. Such continuous culture, in the chemostat, is a situation that reflects bacterial growth in natural environments. In chemostat, growth chamber is connected to a reservoir of sterile medium. Once growth is initiated, fresh medium is continuously supplied from reservoir. 128/226

12 The Chemostat cont d Fluid volume in growth chamber is maintained at constant level by an overflow drain. Fresh medium enters growth chamber at a rate that limits bacterial growth. Bacteria grow at same rate that overflow removes bacterial cells & spent medium. Fresh medium addition rate determines growth rate cos fresh medium contains limiting amount of essential nutrient. Thus, chemostat relieves lack of nutrients, toxic substance accumulation, & accumulation of excess cells in culture, which initiate growth cycle s stationary phase. Bacterial culture can be grown and maintained at relatively constant conditions, depending on nutrient flow rate. 129/226 The Chemostat cont d For Chemostat, model is more complicated than for pure logistic growth with a limited volume of nutrient Eqn.(3.3); as can be seen in Fig. 3.3(b), there is an inflow rate into the central culture chamber and also an outflow rate from the chamber. These latter two terms are both F to conserve mass in chamber. To start, will derive logistic growth model again, for microorganism pop n N, taking into account nutrient conc C(t). 130/226

13 The Chemostat cont d Can define the equations to be: dn = K(C)N = κcn (3.9) dt i.e. bacteria reproducing at a rate C, & change in nutrient conc in culture chamber: dc = α dn = ακcn (3.10) dt dt i.e. α units of nutrient giving one unit of pop n growth. integrating w.r.t. t Eqn.(3.10) get C(t) = αn(t)+c 0, which, on subst n in Eqn.(3.9), gives: dn = κn(c 0 αn) (3.11) dt logistic growth, Eqn.(3.3) for r = κc 0, K = C 0 /α. 131/226 The Chemostat cont d 6WRFN1XWULHQW 5HVHUYRLU & ) 1 & ) %DFWHULDO&XOWXUH&KDPEHU (IIOXHQW (a) (b) Figure 3.3 : Chemostat: Schematic Representations 132/226

14 Chemostat Parameters Quantity Symbol units Nutrient Concentration C mass/volume Reservoir Nutrient Concentration C 0 mass/volume Bacterial population N mass/volume Volume of Growth Chamber V Volume Inflow/ Effluent Flow Rate F Volume/time Table 3.1 : Chemostat Parameters Note from Table 3.1 that by introducing a concentration term into Eqn.(3.9), forced dimensionally to consider N to be the mass of bacteria per unit volume (i.e. a pop n density). 133/226 The Chemostat cont d Modify above chemostat eqns (3.10, 3.11) to account for influx/eflux to/from culture chamber: dn dt = K(C)N FN V where 2nd term on RHS is removal rate of N, & (3.12) dc dt = αk(c)n FC V + FC 0 V (3.13) where 2nd & 3rd terms on RHS are decrease in conc n rate due to removal & increase due to inflow, resp. Volume term V makes eqns dimensionally correct. 134/226