Some mathematical models

The aim of ecological studies is to achieve sufficient understanding of the interactions between organisms and their environments to be able to express these

Figure 5.16 Kelp (Laminaria hyperborea) in (a) spring and summer; the old frond is discarded as the new frond grows and (b) in winter. (c) The tunicate Diazona violacea in winter; only a hard bud remains and (d) in summer.

relationships in precise numerical terms. If this could be done, mathematical models could be constructed from which predictions might be made. The processes regulating phytoplankton production can be formulated in several ways. For example, the rate of change of a phytoplankton stock, P, obviously depends on the rates of addition and loss of cells. Fleming (1939) expressed this relationship as follows:

where a = rate of cell division of phytoplankton, b = initial rate of loss of cells by grazing, and c = rate of increase of grazing intensity. Taking observed values for diatom populations at the beginning and the peak of the spring bloom in the English Channel, he computed curves which fitted observed values well and indicated that, following the peak, the production rate continued to increase despite the fall in phytoplankton stock due to heavy grazing.

Cushing (1959) formulated the rate of change of phytoplankton as:

where P = number of algae (or weight of carbon) per unit volume or beneath unit surface area, r = instantaneous reproductive rate of algae, G = instantaneous mortality of algae due to grazing, and M = instantaneous mortality rate from other causes. r was calculated for observed rates of algal cell division at various light intensities, and account taken of light penetration, compensation depth and depth of wind-mixed layer. In computing grazing rates for observed populations of herbivores, adjustments were made to allow for their reproductive and mortality rates. Applying the equation to the North Sea during the spring bloom, in an area where nutrient depletion was thought not greatly to depress production, Cushing showed fairly good agreement between observed and calculated values for standing stocks of phytoplankton and herbivores.

An alternative to these equations, based on rates of cell reproduction and loss, is a general equation advanced by Riley which takes account of energy considerations, as follows:

where P = phytoplankton population, Ph= rate of photosynthesis, R = plant respiration rate, and G = grazing rate.

Riley (1946, 1947) derived expressions for the coefficients Ph, R and G which take account of illumination, and combined these into the following expanded equation:

P = Phytoplankton population in gC -m~2.

p = Photosynthetic constant, 2.5 (gC produced per gram of phytoplankton C per day per average rate of solar radiation). I0 = Average intensity of surface illumination in gcal-min-1. k = Extinction coefficient (1.7/depth of secchi disc reading in metres).

z1 = Depth of euphotic zone (depth at which light intensity has a value of

0.0015 gcal-cm-2-min-1). N = Reduction in photosynthetic rate due to nutrient depletion,

I-0^5-when concentration is less than 0.55 |.

V = Reduction in photosynthetic rate due to vertical turbulence. R0 = Respiratory rate of phytoplankton at 0°C. r = Rate of change of R with temperature, r being chosen so that R is doubled by a 10°C increase in temperature. T = Temperature in °C.

g = Rate of reduction of phytoplankton by grazing of 1 gC of zooplankton. Z = Quantity of zooplankton in gC X m-2.

From this equation the likely fluctuations of the phytoplankton population over a 12-month period were predicted and showed good agreement with the observed values (Figure 5.17).

An equation was also derived for changes in the population of herbivores, H:

A = Coefficient of assimilation of food R = Coefficient of respiration C = Coefficient of predation by carnivores D = Coefficient of death from other causes.

Despite the many difficulties of evaluation of this equation, predicted and observed values were in substantial agreement (Figure 5.17). For a computer simulation see Steele (1974).

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