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The problem of habitat fragmentation is recently an important issue in ecological research as well as in the practical approach of nature conservation. According to the most popular approaches, habitats are considered as the homogenous parts of the landscape. Also the metapopulation concept problem of the inert habitat heterogenity is considered quite seldom. These approaches have some weak points resulting from the assumption that the border between habitat patches and the metapopulation matrix is fairly sharp. This paper presents a resource-based concept of habitats, based on mathematical theory of point processes, which can be easily applied to analysing the problem of uneven distribution of resources. The basic assumption is that the random distribution of resources may be mathematically described as the realisation of a certain point process. According to our method, it is possible to calculate the expected quantities of available resources as well as the minimum area of habitat that includes the expected abundance of the resource. This approach may be very useful to understand some crucial phenomena in landscape ecology, such as the patch size effect and its connection to habitat loss and fragmentation.

A number of researchers [

In recent two decades, a number of theories have been developed in spatial population dynamics [

Anyway, ecologists and conservation biologists have used many measures of landscape structure, to predict the population dynamics consequences of habitat loss and fragmentation [

Many populations are ultimately limited by resources, and their abundance and distribution will determine the carrying capacity of an environment [

The extrapolation models are not particularly good in dealing with the fact that a random distribution of resources is, by definition, uneven [

The suggested approach is based on the assumption that the random distribution of resources may be mathematically perceived as a realisation of certain point process, e.g. Poisson process. In estimating therefore the intensity of the process, we can learn about the expected occurrence (number, quantity) of a given resource in the selected area. Consequently, we can also calculate the probability that the resource occurs in the given area. Moreover, we can also calculate the probability of the resource occurring in the given area in sufficient quantity to maintain minimum viable populations of certain species. Consequently, one may try to reliably determine the minimum areas required for viable populations. This mathematical tool may be crucial for the process of the locating and size optimisation of protected areas. The methods of estimating the intensity of the Poisson process were described at length by Ripley [

In this paper we would like to present the possibility of using point process in the more general resource- based analysis of the habitat loss mechanism.

Let us denote a certain stochastic environment called state space by

figuration

where

Such a discrete measure

a projection or coordinate function with index B.

Let

Let us now assume that the number and positions of the points may be random. Let

is called a point process on

We know that any set

As we mentioned earlier, point process is a random point measure on the state space

Hence,

The basic point process is a Poisson process. Poisson processes are very convenient building blocks from which to generate other point processes. They can be regarded as the analogue of independent observations and are often called “random” outside mathematics [

Let v be a finite intensity measure on the state space

random variables for each natural

Let us consider a vast open terrain, homogeneous in terms of its characteristics, such as, for example a muddy and dried bed of a shallow lake, and the locations of independently wind propagating seeds of one species selected from those colonising the area. The number of seeds which fall on one square metre of the area is a variable with Poisson distribution, which stems from the fact that there are many such seeds, each with little probability of landing on the marked square [

Let us assume that the distribution of resources is a realisation of a certain Poisson process with the parameter

Obviously, the process does not have to have a constant measure of intensity, as the latter may change, for example, with the lapse of time

In such a case, the probability of the resource occurring

The intensity of a process can also depend on position, which in fact, is commonly seen in biology e.g. we can expect that number of seedlings decrease with distance from the plant producing wind dispersing seeds (

For the sake of simplification, in the following parts of this study, we will consider processes with a constant intensity measure

Let us consider a situation where one resource

By using (7) we can present the relationship between the size of the area

In biological examples, however, the key is often not the occurrence of the resource alone, but also that it occurs in certain quantities. The example of such a situation could be the presence of shoots of a host plant required to maintain the assumed population abundance of an insect [

Hence, when the occurrence of the given species is only possible when the given resource occurs in the defined area a minimum of three times, the probability of PSE occurring is equal to

The relationship between the probability of PSE occurrence and the percentage decrease in habitat area size, for three intensity levels:

Let us have a closer look at the relationship between the intensity of the process and the minimum amount of the resource and their effects on probability of the PSE occurrence (

The theory of point processes may be also applied in the analysis of availability of a number of different resources—which is, in biological terms, much closer to the real situation. Let us consider the simplest situation, where we have two resources _{ }

where

The crucial question at the patch scale is how much habitat must be conserved to ensure the persistence of populations [

Let us assume that the persistence of a certain population in the area

Let us illustrate this with a simple example (

One of the most important aspects of practical nature conservation is to establish protected areas whose objective is to preserve naturally a considerable level of biological diversity, as well as the natural characteristics of ecological processes [

We are grateful to Wojciech Solarz from the Institute of Nature Conservation PAS for his valuable suggestions and comments during preparation of the paper. This study was funded by Institute of Nature Conservation PAS and partly by a Polish State Committee for Scientific Research/National Science Centre grant No. N N304 325836.