Trophic models set out to describe the dynamics of a food web. A number of key terms are described in Table 9.1. The most typical forms of food web models, with applications in terrestrial as well as aquatic ecosystems, are predator-prey models. Predator-prey models (or 'Volterra' models) capture the oscillations that can occur in the abundance of predators and prey (such as foxes and hares). Simple trophic models have been of interest to ecologists because they readily capture behaviour that is like that often seen in real ecosystems. Paradoxically, they have been of interest to mathematicians - in part due to their unpredictable behaviour! To illustrate the principles behind a simple trophic model, the example of a lake ecosystem is considered.

Plot of state variables against time in an enclosed late NPZ P^se plot of an enclosed lake

Plot of state variables against time in an enclosed late NPZ P^se plot of an enclosed lake

Figure 9.1 The results of the trophic-based ecological model of an enclosed lake described in Section 9.2. a) Nitrogen (N), Phytoplankton (P) and Zooplankton (Z) begin at 1 mg.N.m-3. The P is consumed by Z, releasing N back into the lake, which creates an oscillation in concentrations. After a few weeks, the modelled system stabilises around N ~1.5, P ~ 1 and Z ~0.5 mg.N.rn"3. b) The same output in a phase space diagram, showing the attraction of the model to a stable state.

The lake ecosystem is assumed to be made up of three components, N-nitrogen, P-phytoplankton, Z-zooplankton, which will be quantified in mg N m"3 (milligrams of nitrogen per cubic metre). Three processes are considered important: phytoplankton growth, zooplankton grazing and zooplankton mortality. In words, the ecosystem model can then be written:

AN = - uptake for growth of P + regenerated N from Z mortality

where the A symbol represents the change in the value of variable with time. Note that each process appears twice in the equations. For example,

state variables |
a variable whose value changes in time, and describes the state of the system at a given time |

parameter |
a variable whose value is independent of the value of the state variables |

deterministic |
a description of a model or equation in which there is no random, or stochastic behaviour |

stochastic |
a description of a model or equation in which there is a random component |

Figure 9.2 The temperature, dissolved inorganic nitrogen, phytoplankton and zooplankton concentration at depth levels of 0, 33, 66, and 99 m on day 18.5 of a simulation of the effect of northerly winds in the waters off southeast Australia. For more details see Baird et al. (2006). By day 18.5, cool (top left) and nutrient rich (top right) water has been brought to the surface as a result of the upwelling-favourable winds and become entrained in a warm core eddy. A strong phytoplankton bloom develops along the coast at the surface (bottom left). The bloom ends just north of Sydney, being consumed by a zooplankton bloom as water is advected offshore (bottom right). The zooplankton maximum is just downstream of the edge of the phytoplankton bloom.

Figure 9.2 The temperature, dissolved inorganic nitrogen, phytoplankton and zooplankton concentration at depth levels of 0, 33, 66, and 99 m on day 18.5 of a simulation of the effect of northerly winds in the waters off southeast Australia. For more details see Baird et al. (2006). By day 18.5, cool (top left) and nutrient rich (top right) water has been brought to the surface as a result of the upwelling-favourable winds and become entrained in a warm core eddy. A strong phytoplankton bloom develops along the coast at the surface (bottom left). The bloom ends just north of Sydney, being consumed by a zooplankton bloom as water is advected offshore (bottom right). The zooplankton maximum is just downstream of the edge of the phytoplankton bloom.

grazing of P by Z represents a loss of phytoplankton, but a gain in zooplankton. In this way, mass (in this case N) is exchanged between state variables, but total mass of N within the ecosystem does not change. In other words, the modelled system conserves mass.

In order to create a numerical model, the processes must be represented mathematically. A simple representation is given below:

dN/dt = - ^max NP/(k1/2 + N) + mZ dP/dt = + ^max NP/(k1/2 + N) - ^PZ d/dt = + ^PZ - mZ

Note how the equations have a similar form to the 'word' equations given above. Apart from the state variables N, P and Z, and t (time), the equations contain the parameters k^, ^max, ^ and m, which are defined as:

kj/2 half saturation of nutrient uptake 1 mg.N.m-3

^max maximum growth rate of phytoplankton 1 d-1

^ grazing rate coefficient 1 d-1 mg.N-1.m3

m mortality rate of zooplankton 1 d-1

Of course, many other processes, such as phytoplankton mortality or higher order grazing terms, could be considered. The model must now be given initial conditions for each of the state variables. The illustrated simulations will start with initial conditions: N0 = 1 mg.N.m-3; P0 = 1 mg.N.m-3, ZQ = 1 mg.N.m-3. The total amount of mass of the lake is: TN = No + Po + Zo = 3 mg.N.m-3 and, given mass conservation in an enclosed lake, will not change. The equations must be solved forward in time, starting with the initial conditions. Figure 9.1 shows the results from the enclosed lake simulation.

Trophic models are commonly coupled to physical models to capture the effects of advection and mixing on biological quantities. The physical models can range in complexity from simple box models with specified transports to advanced hydrodynamic models that use equations of fluid motion to calculate advection (see Figure 9.2 as an example). This rapidly developing field is considered in depth in a recently published textbook (Williams 2006).

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