One example of applying a population model to evaluate chronic toxicity is from the US Army Corps of Engineers' work with sediment toxicity tests using the estuarine amphipod Leptocheirus plumulosus. Percent survival and fecundity (number of offspring per female) were measured at weekly intervals in 30-week tests of chemically contaminated sediment from Black Rock Harbor (Connecticut, USA). Toxicity was measured in each of the three dilutions of Black Rock Harbor sediment under two food rations to evaluate nutritional effects on toxicity. Three different Black Rock Harbor sediment concentrations were used (0%, 3%, and 6% contaminated sediment in a clean sediment matrix) and two levels of food ration were applied (normal food ration, 1 x; and double the normal food ration, 2 x). Figure 6 illustrates the survivorship and fecundity data obtained from these sediment toxicity tests.

To evaluate the population-level effects of sediment toxicity to Leptocheirus, data for the toxicity test endpoints were entered into the following matrix model:

where n(t) refers to the number of individuals in age class i at time t, Fj represents the fecundity, or birth rate, of individuals in age class i, and Sj refers to the survival rate of individuals in age class i. Note that this is a relatively simple model of a population without density-dependent feedback on vital rates of mortality and birth. The matrix model is applied to successive time steps of the population to project population size and age structure. After several time steps, the population approaches a stable age distribution (i.e., the proportion of individuals in each age class remains constant over time). In this stage, the population grows (or declines) exponentially at a rate determined by the largest eigen value of the population projection matrix:

where Nt = the total population size at time t, N0 = total size at time 0, and A (lambda) = the finite rate of increase.

Relative values of lambda and the change in lambda are useful for summarizing the effects of toxic chemicals or other factors on population growth, despite the fact that real populations in nature rarely are at stable age distribution because of variations in vital rates due to density-dependent processes or other factors (e.g., climatic effects, predation). The relative differences in lambda between treatment comparisons are nonetheless appropriate metrics for evaluating population-level effects.

10 15 Weeks

10 15 Weeks

0% BRH |
- 1X |
3% BRH |
- 1 X |
-e- 6% BRH |
- 1X |

0% BRH |
- 2X |
-■- 3% BRH |
- 2X |
-•- 6% BRH |
- 2X |

»1 |
t |
-1)" |
F2 |
F3 |
F4 .. |
F 1 m |
' »1 (t )" | |||||

»2 |
t H |
-1) |
S1 |
0 |
0 |
0 .. |
0 |
»2 (t) |
œ |
15 | ||

es | ||||||||||||

»3 |
t H |
-1) |
0 |
S2 |
0 |
0 .. |
0 |
»3 (t) |
E | |||

»4 |
t H |
-1) |
0 |
0 |
S3 |
0 .. |
0 |
»4 (t ) |
c |
10 | ||

Ci | ||||||||||||

0 |
.2 | |||||||||||

O |
5 | |||||||||||

_»m |
(t - |
_ 0 |
0 |
0 |
0 Sffl— |
1 Sm |
_ »m(t) _ |

Figure 6 Survivorship and fecundity curves for Leptocheirus plumulosus during exposure to Black Rock Harbor (BRH) sediment under six combinations of different sediment dilutions (%) and food rations (X).

10 15 Weeks

Figure 6 Survivorship and fecundity curves for Leptocheirus plumulosus during exposure to Black Rock Harbor (BRH) sediment under six combinations of different sediment dilutions (%) and food rations (X).

The largest contributions to differences (or changes) in lambda can be determined by using the following equation:

where a.y = the element in the ith row and the jth column of the population projection matrix for either the experiment treatment (e) or control (c). Each term in the summation represents a contribution by a certain age-specific survivorship or fecundity term to the change in lambda. This approach was used to infer that the largest contributions to the difference in lambda between experimental treatments for Leptocheirus were generally due to changes in survival rates of age classes 1-3 and fecundities of age classes 4-8. As expected, increasing sediment concentration in the Black Rock Harbor sediment tests resulted in decreases in lambda and increasing food resulted in increasing lambda (Table 4).

Control |
Treatment |
Change in lambda |

(0%, 1 x) |
(3%, 1x) |
-0.071 |

(0%, 2x) |
(3%, 2x) |
-0.092 |

(0%, 1 x) |
(6%, 1x) |
-0.221 |

(0%, 2x) |
(6%, 2x) |
-0.227 |

(3%, 1 x) |
(6%, 1x) |
-0.150 |

(3%, 2x) |
(6%, 2x) |
-0.135 |

(0%, 1 x) |
(0%, 2x) |
-0.081 |

(3%, 1 x) |
(3%, 2x) |
-0.060 |

(6%, 1 x) |
(6%, 2x) |
-0.075 |

In the model presented above, vital rates and population growth are density independent. In the more general case, a density-dependent model would be used. Population models with density dependence, either as a matrix formulation or a continuous process model (e.g., using differential equations), have been developed for burrowing mayfly (Hexagenia spp.) in Lake Erie (USA), Japanese crucian carp (Carassiun auratus) in Lake Biwa (Japan), fathead minnow (Pimephales promelas) in the laboratory, song sparrow (Melospiza melodia) in the Mandarte Islands (Canada), harp seal (Phoca groenlandica) in the northwest Atlantic, and many other species. Metapopulation models are a special class of models that incorporate density-dependent processes by modeling a set of populations of the same species in the same general geographic area and including migration among populations.

Not all of the model applications discussed above were used for ecotoxicological assessments, but the models could be easily adapted for this purpose. In general, to apply a population model to a chemical risk assessment, the following data are needed:

• estimates of demographic parameters in absence of stress;

• toxicity data for selected endpoints, for example, (1) survivorship, (2) fecundity, (3) growth;

• estimate of parameters for density-dependent modeling, such as intrinsic rate of natural increase (rmax) and carrying capacity (K) or functions relating vital rates to density;

• initial age or stage structure for models of structured populations; and

• locations of suitable habitat patches, initial organism distribution (or presence-absence data), dispersal rates, and possibly time series of the distribution of habitat types for spatially explicit models, such as metapopulation models.

Examples of variables incorporated into population models are provided in Tables 1-3. Graphic forms of some commonly used population models are shown in

Figure 7, including the Malthusian exponential growth model, the Malthusian model with uncertainty analysis (e.g., Monte Carlo), the density-dependent logistic growth model, and various spawner-recruit models that are popular in fisheries management.

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