The carrying capacity _{t+1} = _{t} + _{t}, in which _{t} is the population densities at time _{t+1} is the density at a subsequent time _{t} = _{m} − _{t}, in which _{t} is the per capita rate of increase at time _{m} is the maximum possible individual rate of increase, and the negative slope _{t} = 0, so that _{t} = _{m} – _{m}/_{max}) or unfavourable (_{min}). Values of _{max} and _{min} are estimated for

Early work by

Foraminiferal micropalaeontologists have acquired a considerable number of time-series (e.g.

A change in the carrying capacity of an impacted area indicates that its economic development is having a negative impact on the ecoystem. There is thus a need for a simple method for determining

A population is a group of conspecific individuals that exist together in time and space (

It follows that a given population of _{t} at time _{t+1} as follows:

Thus, Δ_{t+1} − _{t} and may be positive (population growth), zero (population stable) or negative (population decline), depending on the balance of inputs and outputs.

For Δ_{t} per spatial unit (area, volume) is more informative than is a census of an unconstrained population and is termed the population density. The expression Δ_{t+1} − _{t} per spatial unit measures the change in population density between time _{t} and _{t+1} signify population densities.

Whether the density of a studied population increases, is static or decreases between times

As Berryman & Kindlmann (

algebraic manipulation of which gives

in which the population density at time

For a species to survive indefinitely at a site, the positive feedback loop in _{m}. As the population density increases, (

in which _{t} is the per capita rate of increase at time _{m} is the maximum possible individual rate of increase, _{t} is the population density at time _{t} against _{t} indicates not only _{m}, which occurs when the population density approaches zero (i.e. at the intercept on the y-axis), but also a population density _{t} = 0 (^{2}, in which 0 < ^{2} < 1. This coefficient indicates the strength of the linear association and the percentage of the data close to the line of best fit. If, for example, ^{2} = 0.75, then 75% of the total variation in _{t} and _{t}. The remaining 25% of the variation in _{t} and _{t} are sufficient to explain the values of _{m} and ^{2} and significant results from ANOVA are obtained by graphing _{t} = _{m} − _{t} at the lag of the feedback mechanism, whether it is _{t+2}, _{t+3}, etc. (

Graph showing the determination of the maximum per capita rate of population density increase (_{m}) and carrying capacity _{t} = _{m} − _{t} where _{t} measured the per capita rate or reproduction at time _{t} may be positive, zero or negative.

Setting _{t} = 0 in _{t} = _{m} –

Thus, K can be readily determined after conducting the linear regression in _{m} and

Aseasonal environments are likely to be rare. For example,

Population dynamics in a phenological environment constitute chaotic systems in which it is not possible to predict precisely the population density for any month in successive years (_{max}) at that time of year when the resource supply in the environment is favourable and low (_{min}) when it is not (see also _{max} and _{min} act as attractors comparable in form to the Lorenz chaotic attractor (cf. _{t}, _{t} and environmental favourability cycling between _{max} and _{min} over time. To calculate _{max} and _{min}, a seasonally variable time-series must, therefore, be split into two subsets of data: those occasions (weeks, months) that contribute towards _{min} and those that contribute to _{max}. It is inadvisable to partition the time-series using samples with more or less than the mean population density as the mean is influenced by outliers (_{max}. The time-series is instead partitioned here using the median population density. Also, given that the population _{t+1} shows density dependence on the population density _{t}, it is not possible simply to splice together (albeit in order of time) all population density readings either above or below the median and analyse these as a single time-series using ^{2} because _{t} = _{m} − _{t} is not applicable across the splices. Instead, the longest continuous sets of readings above and below the median must be analysed separately to give estimates of _{max} and _{min}, respectively. This method is demonstrated here using time-series from

Three time-series are analysed here, each for a different reason.

_{t} of live _{t} was greater or lower than the median. For the longest of these shorter times-series the monthly values of per capita _{t} [=(_{t+1} − _{t})/_{t}] were calculated and linearly regressed against _{t} to give _{t} = _{m} − _{t}, and the carrying capacity _{min} and _{max} derived from _{m}/

Several measures not determined by ^{2}) of the linear regression was determined, as were the 95% confidence limits for _{m} and _{t} and _{t} were sufficient to explain the values of _{m} and _{m} and _{min} and _{max}.

The monthly population density of _{t+1} against _{t} returned _{t} = 0.920 – 0.008_{t} (^{2} = 0.493). This equates to a _{min} of 115 _{t} and _{t} (F_{(1,2)} = 1.943, _{m} and _{min} of –72 to 251 _{min} is taken as ranging from 0–251

Time-series of population densities _{t} of _{min}) and maximum (_{max}) carrying capacities are superimposed across the entire time-series.

Time-subseries for _{t}) in the Exe Estuary, UK: (

Date | _{t} |
_{t} |
---|---|---|

(_{t} < median |
||

March 1981 | 164 | −0.35976 |

April 1981 | 105 | 0.31429 |

May 1981 | 138 | −0.17391 |

June 1981 | 114 | −0.28070 |

July 1981 | 82 | n\a |

(_{t} > median |
||

September 1980 | 341 | 0.20235 |

October 1980 | 410 | −0.36098 |

November 1980 | 262 | 0.35115 |

December 1980 | 354 | −0.17514 |

January 1981 | 292 | −0.04110 |

February 1981 | 280 | n\a |

n\a, not applicable.

Intercept _{m} and slope _{t} = _{m} − _{t} and estimated minimum (_{min}) and maximum (_{max}) carrying capacities for time-subseries for selected foraminifera in seasonal environments: (_{min} 1–4 = four subseries of four samples each); (_{max} calculated at lags of _{t+1} and _{t+3}.

Time-series | Carrying capacity | Measure | Value | Lower bound (95%) | Upper bound (95%) |
---|---|---|---|---|---|

( |
_{min} |
_{m} |
0.921 | −2.356 | 4.198 |

−0.008 | −0.033 | 0.017 | |||

_{min} |
115 | −72 | 251 | ||

_{max} |
_{m} |
1.336 | −0.465 | 3.137 | |

−0.004 | −0.009 | 0.001 | |||

_{max} |
331 | −49 | 2368 | ||

( |
_{min} 1 |
_{m} |
0.48 |
−7.064 |
8.025 |

_{min} 1 |
10487 | −184 | 210 | ||

_{min} 2 |
_{m} |
0.509 | −15.056 | 16.074 | |

0.0012 | −0.033 | 0.030 | |||

_{min} 2 |
436 | −462 | 531 | ||

_{min} 3 |
_{m} |
1.348 | −11.852 | 14.549 | |

−0.005 | −0.053 | 0.043 | |||

_{min} 3 |
262 | −222 | 338 | ||

_{min} 4 |
_{m} |
2.153 | 0.677 | 3.629 | |

−0.005 | −0.009 | −0.001 | |||

_{min} 4 |
412 | 73 | 3202 | ||

_{max} |
_{m} |
1.015 | −0.379 | 2.409 | |

−0.001 | −0.001 | 0.000 | |||

_{max} |
2012 | −300 | 9457 | ||

( |
_{min} |
_{m} |
2.017 | 1.78 | 2.252 |

−0.0051 | −0.00572 | −0.00442 | |||

_{min} |
398 | 312 | 510 | ||

_{max} at _{t+1} |
_{m} |
0.231 | −0.738 | 1.201 | |

−0.0002 | −0.00095 | 0.00055 | |||

_{max} |
1180 | −781 | 2170 | ||

_{max} at _{t+3} |
_{m} |
1.735 | 0.358 | 3.112 | |

−0.0013 | −0.0023 | −0.00023 | |||

_{max} |
1368 | 155 | 13 317 |

The longest continual time-series for _{t+1} against _{t} returned _{t} = 1.336 – 0.004_{t} (^{2} = 0.659), which indicates _{max} of 334 _{t} and _{t} (_{(1,3)} = 5.744, _{m} and _{max} lies somewhere between 0 and 2368

The monthly population densities of _{min} of 262–10 487 _{t} and _{t} for only for one of these subseries (January–April 1996; _{(1,1)} = 263.2, _{m} and _{min} to range from 0–10 487 specimens per 80 ml (

Time-series of population densities _{t} of _{min}) and the one of maximum (_{max}) carrying capacities are superimposed only in the vicinity of the samples used to determine them. Population densities have been plotted as natural logarithms due to the wide range of values.

Time-subseries for _{t}) in the Indian River Lagoon, Florida, USA: (

Date | _{t} |
_{t} |
---|---|---|

(_{t} < median (1) |
||

February 1992 | 126 | 0.56349 |

March 1992 | 197 | 0.25888 |

April 1992 | 248 | 0.59274 |

May 1992 | 395 | n\a |

(_{t} < median (2) |
||

December 1993 | 494 | 0.13765 |

January 1994 | 562 | −0.25623 |

February 1994 | 418 | −0.07656 |

March 1994 | 386 | n\a |

(_{t} < median (3) |
||

December 1994 | 385 | −0.35065 |

January 1995 | 250 | −0.51200 |

February 1995 | 122 | 1.01639 |

March 1995 | 246 | n\a |

(_{t} < median (4) |
||

January 1996 | 176 | 1.22159 |

February 1996 | 391 | 0.16113 |

March 1996 | 454 | −0.25991 |

April 1996 | 336 | n\a |

(_{t} > median |
||

May 1996 | 985 | 0.64873 |

June 1996 | 1624 | −0.07081 |

July 1996 | 1509 | 0.83830 |

August 1996 | 2774 | −0.17628 |

September 1996 | 2285 | −0.47921 |

October 1996 | 1190 | 0.09916 |

November 1996 | 1308 | n\a |

n\a, not applicable.

The longest subseries for _{t} = 1.015 – 0.001_{t} (^{2} = 0.46), which equates to a _{max} of 2011 _{t} and _{t} (_{(1,4)} = 3.40; _{m} and _{max} probably lies between 0 and 9457

Of the 25 samples in this time-series, the longest subseries with lower than the median population density of 528 specimens per 130 ml of sediment contained five samples only (18 February–4 April 1996; _{t} against _{t} gave _{t} = 2.017 – 0.005_{t} (^{2} = 0.998), indicative of a _{min} of 398 _{t} and _{t} (_{(1,2)} = 1130.8, _{m} and _{min} probably lies between 312 and 510

Time-subseries for _{t}) in lower Cowpen Marsh, UK: (_{t} calculated using _{t} and _{t+1}; (_{t} calculated using _{t} and _{t+3}.

Date | _{t} |
_{t} |
---|---|---|

(_{t} < median |
||

18 February 1996 | 216 | 0.925926 |

5 March 1996 | 416 | −0.07692 |

19 March 1996 | 384 | 0.041667 |

4 April 1996 | 400 | 0 |

17 April 1996 | 400 | n\a |

(_{t} > median ( |
||

14 May 1995 | 904 | 0.212389 |

1 June 1995 | 1096 | −0.06204 |

12 June 1995 | 1028 | 0.093385 |

28 June 1995 | 1124 | 0.346975 |

12 July 1995 | 1514 | 0.321004 |

27 July 1995 | 2000 | −0.28 |

10 August 1995 | 1440 | −0.42222 |

26 August 1995 | 832 | −0.30769 |

9 September 1995 | 576 | n\a |

(_{t} > median ( |
||

14 May 1995 | 904 | 0.243363 |

1 June 1995 | 1096 | 0.381387 |

12 June 1995 | 1028 | 0.945525 |

28 June 1995 | 1124 | 0.281139 |

12 July 1995 | 1514 | −0.45046 |

27 July 1995 | 2000 | −0.712 |

10 August 1995 | 1440 | n\a |

26 August 1995 | 832 | n\a |

9 September 1995 | 576 | n\a |

n\a, not applicable.

Determining the maximum and minimum carrying capacities for _{t} = _{m} − _{t} for a subseries of four samples with a lower than median population density _{t}. (_{t} = _{m} − _{t} for a subseries of eight samples with a greater than median population density _{t} at a lag of _{t} of _{min}) and maximum (_{max}) carrying capacities are superimposed only in the vicinity of the samples used to determine them.

The longest subseries with a greater than median density consisted of nine samples (14 May–9 September 1995) for which linear regression of _{t} against _{t} at a lag of _{t} = 0.231– 0 .00019_{t} (^{2} = 0.064). The graph of _{t} against _{t} at a lag of _{t} against _{t} at a lag of _{t} = 1.735 – 0.0013_{t}, ^{2} = 0.743, _{t} and _{t} (_{(1,4)} = 11.59. _{m} and _{max} lay somewhere between 156 and 13 317

Interpretation of population density time-series from foraminifera in the wild may, however, be made problematic by biotic interactions. For example,

This paper examined three foraminiferal time-series (

No matter what their length, the three foraminiferal time-series contained only short subseries with population densities greater or lower than the median. Linear regression of _{t} against _{t} for these short subseries for _{min} and _{max} reliably or distinguish between them, there being much overlap of the 95% confidence intervals for _{m} and _{min} for _{max}. Murray (^{2}). Thus, the observed densities of _{max} (2368 individuals). This probably arose from the time-subseries containing so few samples that the signal was temporally aliased (cf. _{t} adequately (cf. _{t} and _{t} and high values of ^{2}, despite containing about the same number of samples as the subseries for _{min} and _{max} for

If _{min} and _{max} for foraminiferal population densities are to be used in environmental monitoring, then some means must be found to increase the values of ^{2} and narrow the confidence intervals for _{m} and ^{2} and wider confidence intervals for _{m} and _{m}, _{min} and _{max}. Instead, it seems from analysis of

Calculating _{m} and the strength of interaction between individuals in the population _{m} and

The concept of the carrying capacity _{min} and _{max}, respectively) can be readily calculated from time-series of foraminiferal population densities. However, the precision with which they can be calculated depends on the quality of the time series. Further work is required to see if the quality is improved by collecting a large number of replicates or frequent (weekly, fortnightly) samples of relatively few replicates. Armed with this knowledge, foraminiferal workers will be able to contribute significantly to environmental impact assessments. They will, in particular, be able to detect recovery (or lack thereof) from adverse environmental impacts much more quickly than those working with longer-lived organisms.

A financial contribution from the Research and Publications Fund of the University of the West Indies is gratefully acknowledged. Thanks are due to John W. Murray and an anonymous reviewer for their valuable and constructive comments.

Scientific Editing by Alan Lord