Title : Total mortality of fish due to starvation: Catastrophe of the fold
Abstract:
A model describing growth and death of ectothermic animals under unfavorable temperature and nutritional conditions of the external environment is proposed. The model is based on the balance equation of growth dW/dt=A-T, where W is the body weight, t is the time (age), A is the assimilation rate, and T is the expenditure of general metabolism.
Symbol β denotes the effective concentration of available food in the environment, θ is the environmental temperature, a, b, c, γ =const. Equations (1) are derived from the model of nutrition (Rashevsky, 1959) and from some relations of dimensional analysis. Expression exp(γ θ) describes the dependence of the rates of biochemical reactions on temperature according to the Vant-Goff rule. According to our model, death of individuals over lack of food occurs at the moment when the inequality , becomes strict equality. This death condition describes the critical intensity of irreplaceable protein breakdown in enzyme systems during starvation. Let us introducing new variables Variable F will be named here as temperature-food environmental factor. Then the relationship for the equilibrium conditions can be written as the quadratic polynomial.
The model parameters were estimated from Melnichuk (1975) and Ivlev (1955) data characterizing feeding, growth and starvation mortality in 15 species of juvenile freshwater fishes. The parameter values averaged over all species were as follows: a=0.400±0.056 m2/(gram·day), b=17.3±3.5 m2/gram4/3, c=0.0120±0.0013 gram1/6/day, K= (9.89±1.93) ·10-4 /day, γ =0.100±0.000 (deg C)-1. Dimensions of the model variables are the next: [W] = gram, [β] = gram/(m2 or m3), [θ] = degrees C, [t] = day.
The figure schematically depicts the phase portrait of the system describing the dynamics of average body weight W1/6 in even-aged cohort of animals and the dynamics of the factor F. The arrows indicate various directions of the system movement. The horseshoe curve is the quadratic polynomial (2) for equilibrium. This curve is called the fold in the mathematical theory of catastrophes (Gilmore, 1981). Its upper branch (solid curve) describes the relationship
between the Lyapunov’s stable definitive body weight of adults in a population and the temperature-food factor F. The trend described by the upper branch of the fold reflects the well-known biogeographic rule of Bergman: body sizes are smaller in low latitudes with a warm climate. The pattern of the lower branch of the fold suggests that small immature individuals in a population are more sensitive to lack of food and to high temperature. They are more likely to starve to death than larger organisms under these conditions. It follows from the model that population inevitably dies out regardless of the body size of individuals at F smaller than critical (bifurcation) value equal to 4b(c/a)2. This critical value is marked with the asterisk in abscissa axe of the figure. It represents the value of the factor F where upper and lower branches of the fold merge together.
Further development of the model makes it possible to describe auto-oscillations in the population dynamics and body size structure under conditions of insufficiently abundant food supply and elevated environmental temperature. These auto-oscillations are caused by the threshold character of "switching on" (F<*) and "switching off" (F>*) the trigger mechanism of total mortality of individuals due to starvation. The next problem requires the use of a more complex temperature dependence for the rate of biochemical reactions instead of the simple Vant-Goff rule. Enzymes are denatured and the metabolic rate drops sharply at excessively high temperature. This thing should lead to a more complex phase portrait of ectothermic animal growth: closed ellipse appears instead fold catastrophe.
