GROWTH MODELS

 

An accurate growth estimate is fundamental to any assessment of population demographics.  In this simulation you will use a simplified version of the von Bertalanffy growth model along with real data to predict the growth of blue crabs.

Growth model:        Differential equation that allows us to mathematically model the growth of a population continuously with time.

 

Growth rate:            an indication of how fast a population can reach its maximum possible size.

 

Von Bertalanffy growth model:            L(t) =  Lº*(1-e(k))(t-t0)  

                This model was introduced by Von Bertalanffy in 1938 and is widely used in the fish industry

 

Variables:

 

Lº = maximum possible size

L is determined by measuring the maximum size of crabs known to be very old( > 3years)

K = rate of growth

K is determined by rearing a sample population in which growth and age can be accurately measured.

T = Temperature degree day

·       We will use something called temperature degree days instead of calendar days.

·       The temperature º day takes into account that crabs are will only grow on days above 9º C and also grow more rapidly as water temperatures rise.  Plotting growth vs. time would be misleading since it would not take temperature into account. 

·       Temperature degree days are calculated by adding the number of degrees above 9 ºC for that day to the preceding temperature degree day, for the length of 1 calendar year.

 

Note: Temperature º days do not directly correspond to calendar days since there can be a 0 ºdays  for example the day 5/12/03 in the example below.

 

Ex. 

Date

Temperature

(C)

Temperature º days

5/12/03

8

0

5/13/03

10

1

5/14/03

14

6

5/15/03

5

6

5/16/03

10

7

5/17/03

14

12

 

Procedure:     Using growth models to quantitatively measure crab growth

  1. After reading the directions, click on choptank water temperature.  It is an excel spreadsheet containing actual surface water temperature for the choptank river for the year 2000. (http://www.cbos.org/)  

2.  This data will be used to complete several additional calculations.

3.  Use a separate function for each additional column.

Use a function to Sum the total number of  º days for the year 2000.

Using the function $H$2*(1-$I$2*EXP(-($J$2*E3))) calculate expected carapace width

  Note: L and K are constants in this equation, however they could change based on the experimenters data                   

Lº       determined to be 240 mm using data from commercial fisherman

K         determined to be .0004 by observing the growth rate of pond raised blue crabs

 

7.  Graph and label growth curve using columns E & F

              X – axis = º days             Y- axis = carapace width

 

Questions:

 

  1. Based on the growth model, what is the maximum carapace width that you would expect a juvenile crab to reach in the year 2001?

   

  1. 1988 is considered a cold year with low water temperatures and 2500 ° days, much lower than 2001. 

Using the growth model and the number of °days, what would you expect the maximum carapace width to be for a juvenile crab growing in 1988?

 

  1. What problems for the crab population arise by setting fishing thresholds solely on carapace width?

   

  1. What variables should a fishery consider when trying to set sustainable threshold limits?

 

 

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