GROWTH IN LENGTH: a model for the growth of an individual

Transcription

1 GROWTH IN LENGTH: a model for the growth of an individual Consider the following differential equation: L' L What are the dimensions for parameters and? As was the case with the differential equation relating N' with N, once again there is a linear relationship between L' and L. In this new differential equation is the intercept of the line and is the slope. The Differential Equation. Solution to the Differential Equation. L / 0 0 t The graph on the left defines the slope for each point of the curve on the right. Analytical Solution to / - L As was the case for the differential equation N' ( t) M N( t), we can use an algebraic approach to drive an analytical solution for the differential equation for growth-in-length. L L is the dependent variable, t is the independent variable. L Separate the variables by putting L on one side and t on the other. FW431/531 Copyright 2008 by David B. Sampson Growth - Page 11

2 1 L Integrate both sides to "undo" the differentiation. Remember the Rule F( x) x f( u) du <> a d dx F( x) f( x) If we differentiate F(x), the function that the integral is equal to, we should recover the function f(x) that is under the integral sign. Start with the right hand side: t + C The derivative of [- t + C] is -. Then do the left hand side: 1 L dx /(X + a) ln(x + a) + constant ln L + C' The derivative of [ln(x+a) + C'] is dx/(x+a) + C'. The handout "Review of Some Mathematics" has tables of integrals that you can use. Wikipedia has extensive lists. Finding integrals can be hard. Now combine the two sides: ln L t + C'' C'' combines the earlier arbitraty constants, C'' C - C' L( t) C''' exp + ( t) Exponentiate both sides to get the general solution, C''' exp(c''). Define a new parameter L inf /. and substitute it for /. L( t) L inf + C''' exp( t) Specify the initial conditions, L0 at tt 0, and solve for the arbitrary constant C'''. L( t 0 ) 0 L inf + C''' exp t 0 > L inf C''' exp t 0 L inf C''' exp( t 0 ) > C''' L inf exp t 0 The particular solution is L( t) L inf L inf exp t 0 exp( t) L( t) L inf 1 exp FW431/531 Copyright 2008 by David B. Sampson Growth - Page 12

3 This equation in fisheries science is usually known as the von Bertalanffy growth equation and it is widely used as a model for the growth in length of individual fish. However, see night (1968) and Roff (1980) on the Supplemental Reading list for dissenting opinions about the utility of the von Bertalanffy equation. Ricker (1979) on the Recommended Reading list provides a comprehensive review of the numerous other growth models that have been proposed. Note that the parameter t 0 is a mathematical artifact that arises from the arbitrary constant in the differential equation. It does not represent the age at conception, although some authors have tried to use it as such. Explore the effects of Linf,, and t 0 using the Excel demonstration. Alternative Derivation of the von Bertalanffy Model The von Bertalanffy growth model can also be formulated as a difference equation, which is similar to a differential equation but involves differences ( X) rather than differentials (dx). The difference equation form of the model provides a different view of what it means for a fish to grow according to the von Bertalanffy equation. We can derive the difference equation model from the solution, L(t). L inf L inf exp L t + t ( t t + t 0 ) Length at time t + t. L( t) L inf L inf exp t + t 0 Length at time t L t L t + t Do the same derivation for L(t) - L(t t). ( 1) L inf exp t + t 0 exp t The change in length. L( t) L inf L inf exp t + t 0 Length at time t. L( t t) L inf L inf exp( t + t + t 0 ) Length at time t - t L( t) L t t Now take the ratio of the two differences. L( t) L( t t) L t + t L t L inf exp t + t 0 1 exp t The change in length. 1 exp t exp t The last term is just a 1 exp( t) exp( t) multiplication by one. 1 exp( t) ( 1 exp( t) ) exp( t) 1 exp t exp t We can rearrange this result to derive the difference equation L t L t + t exp t L t ( L( t) L( t t) ) L t+ t exp t The subscripts indicate that the L values are from different time periods. Change in length ( L) is the same as growth. So, we can express the equation as FW431/531 Copyright 2008 by David B. Sampson Growth - Page 13

4 Growth t Growth exp t t+ t Note that 0 < exp(- t) < 1 for 0 < and 0 < t. In words this equation states that the growth increment during each time-step is a constant fraction of the growth increment during the previous time-step. Here is a graph that illustrates the idea. If ( t) is near zero, then growth is almost linear because exp(0)1 so that L t+1 L t. L 2 L inf If ( t) is large, then there is rapid decline in L because the exponential term will be close to zero; L(t) will rapidly approach L inf. Length L 0 t L 1 t For alternative forms for the von Bertalanffy equation and their statistical properties see Ratkowsky (1986) on the Supplemental Reading list. If you are trying to fit a given non-linear model to data using non-linear least-squares regression, some forms of the model may produce "better" results than others in that the parameter estimates will be more closely distributed as normal random variates rather than being highly skewed. Weight of an Individual Fish We can build a model for the weight of a fish using the following ideas. Mass Density Volume The units are [ M ] [ M / L 3 ] [ L 3 ]. If an animal when it grows does so without changing either its shape or its density, then this animal is said to be growing isometrically. If the animal does not change its shape as it grows, then its volume is proportional to any linear measure of its size. Isometric Growth Volume a Linear_Measure 3 The coefficient a is a constant of proportionality. Weight a' Length 3 a' is just the density. FW431/531 Copyright 2008 by David B. Sampson Growth - Page 14

5 Now combine this with the von Bertalanffy growth equation for length. W a' L 3 and L( t) L inf 1 exp 3 W( t) a' L inf 1 exp W( t) W inf 1 exp 3 3 where W inf a' L inf What are the dimensions for W inf,, and t 0? Explore the effects of W inf,, and t 0 using the Excel demonstration. Condition Factors Sometimes researchers use condition factors to compare different samples of fish. Condition_Factor W L 3 Provided growth is isometric, the condition factor accounts for differences in weight that are due to differences in length. Heavier fish will have larger condition factors. It is probably not appropriate to use condition factors to compare individuals from different species because they may have fundamentally different growth processes. Allometric Growth What do we do if the fish do not grow isometrically? Suppose we have the following relationship between weight and length. W a L b and b 3 Fish may change shape or density as they grow, in which case their growth is not isometric. For most fish species the length-weight parameter b is approximately equal to 3. Individual Growth in Weight If we apply the allometric growth to the von Bertalanffy equation for growth in length we get a general equation for growth in weight. b W( t) W inf 1 exp Biomass of a Cohort A cohort is an identifiable group of fish. Usually the term refers to all those fish in a stock that were born during the same year and are therefore the same age. The term stock refers to a self-sustaining reproductive population. Cushing (1968), on the Supplemental Reading list, discusses some of the issues with defining a unit stock. Given that most populations have some mixing with other populations, it is problematic to come up with a precise definition. Later in the course we will explore a model of a fishery operating on a mix of stocks. Most of the models we will consider, however, assume that the population is a single stock whose individuals do not mix with FW431/531 Copyright 2008 by David B. Sampson Growth - Page 15

6 others. When spawning occurs throughout the year, defining age is problematic. For example, fish ages are often assigned on a calendar year basis, but the fish born on an. 1st of a given year are not one year younger than the fish born on Dec. 31st of the previous year. We will ignore this problem in the models we will develop. Suppose we have a cohort of identical fish, all born simultaneously; e.g., a stocked fish pond. We will take the model for survival and the model for growth in weight and build a simple model showing how biomass changes. Numbers at Age: N( t) N( 0) exp( M t) If we define a cohort based on age, then age and time measure the same thing. Weight at Age (isometric growth): Cohort Biomass: 3 W( t) W inf 1 exp B( t) N( t) W( t) Here is a numerical example. N W inf 4.0 M { 70% survival per time step } 0.5 { 60.7% growth increment per time step} t N( t) N0 exp( M t) W( t) W inf 1 exp B( t) N( t) W( t) t N( t) W( t) B( t) On the next page are the graphs of N(t), W(t), and B(t). FW431/531 Copyright 2008 by David B. Sampson Growth - Page 16

7 N(t) 4 W(t) B(t) The biomass graph clearly has a maximum value. If you wait too long to harvest, you lose biomass to mortality. If you do not wait long enough, you lose potential growth One might well ask at what age is the biomass a maximum? From calculus you may recall that the maximum (or minimum) occurs at that value of t for which the slope B'(t) is zero. B( t) N( t) W( t) One of the basic rules for taking derivatives: d[ X Y ] d[x] Y + X d[y] db N dw + W dn 0 > N dw W dn Depending on the equations for N and W, we may not be able to solve this equation analytically, but we can always solve for t numerically. However, it is even simpler to just read the value off the graph. If you are raising fish to make money, then the age at which you should harvest the fish will probably not be the age at which their biomass is a maximum. For example, suppose you pay someone to feed and care for the fish, and you want to maximize your profits ( Revenues - Costs ). FW431/531 Copyright 2008 by David B. Sampson Growth - Page 17

8 Profits here are defined as Π ( t) Price B( t) Wages t Wages t Maximum (or minimum) profits occur when Price B( t) dπ Price db 0 Wages $$ t The maximum profits occur where the revenue and costs curves have the same slopes. But notice that there are two points where this is true. At the other point (with the smaller t) the profits are at a minimum (and losses are at a maximum). FW431/531 Copyright 2008 by David B. Sampson Growth - Page 18