We consider trophic networks, a kind of networks used in ecology to represent feeding interactions (what-eats-what) in an ecosystem. We observe that trophic networks can be naturally modelled as Petri nets and this suggests the possibility of exploiting Petri nets for the analysis and simulation of trophic networks. Some preliminary steps in this directions and some ideas for future development are presented.
Ecosystems are very complex systems constituted by biotic communities (populations of di erent species), abiotic components of the environment (like air, water, soil) and interactions among these (living and non-living) elements. A branch of ecology deals with the study of feeding relationships within ecosystems and represents them as networks of interacting compartments called trophic networks or food webs. Due to the common limited availability of experimental information, a static approach (the mass balance steady state approach) to the study of such networks has been developed as alternative to the dynamic description.
Complex networks of interacting entities are widely studied in computer science: computer networks, agent systems, and, in general, all concurrent and distributed systems fall into this category. Uncountably many formalisms and practical tools have been developed for the representation and analysis of interacting systems. This suggests the possibility of reusing models and techniques from computer science for the study of trophic networks.
This idea is pursued in [
In this paper we explore the use of another widely used model of
concurrency, namely Petri nets [
The structure of the paper is as follows. In Section 2 trophic networks are introduced with a small case study related to the Venice lagoon. In Section 3 the main concepts in Petri nets used to model trophic networks are brie y recalled. In Section 4 we propose a simple application of Petri nets to the representation and analysis of trophic networks when dynamic information are not available. This is exempli ed in the case study. Some conclusions and suggestions for further work are given in Section 5. 2
An ecosystem is a community of living organisms, such as plants, animals and microbes, in conjunction with the nonliving components of their environment, such as air, water and bioavailable organic matter (detritus), which interact as a system. A trophic network (or food web) is a representation of feeding interactions in an ecosystem, where the components are connected by binary links (what-eats-what). Food webs permit to represent and analyse the trophic structure and functioning of an ecosystem. This knowledge can be used to identify key species and to detect anthropogenic impacts, such as the e ects of pollution, of physical disturbance, of resources exploitation, etc. Real trophic networks are very complex, hence models provide partial and abstract representations where, for instance, similar species are aggregated into groups with similar feeding behaviour. Model representation of a trophic network generally focuses on the uxes of energy or biomass between nodes. Such uxes are directional and generally encompass some very relevant organism-level processes, such as production, consumption, assimilation, predation, non-predatory mortality and respiration. An ecosystem is generally an open system, i.e. there are ows of material or energy between the system and the rest of the world. For this reason, when representing and analysing trophic networks, generally also the input and output ows are taken into account. Inputs can be primary production, immigration or incoming of detrital matter into the system, while outputs can be emigration, harvesting by humans and exit of detrital matter from the system. Some energy may be dissipated into heat (respiration) or some material may be degraded into its lowest energy form (detritus).
Knowledge on the species present in the studied ecosystem and on their
feeding behaviour is a needed prerequisite for representing the trophic network. First
of all it is necessary to single out the n living and non-living compartments to be
represented. A compartment can represent a population of a given species or of
some aggregation of species with comparable feeding habits. For each
compartment it is necessary to determine which other taxa are included in its diet, thus
Nocost. Flux
1 CO2!PHP
2 input!DET
3 PHP!MIZ
4 PHP!MEZ
5 PHP!DET
6 PHP!TAP
7 DET!BPL
8 BPL!CO2
9 BPL!MEZ
10 BPL!MIZ
11 BPL!TAP
12 MIZ!MIZ
13 MIZ!DET
14 MIZ!CO2
15 MIZ!MEZ
16 MIZ!TAP
17 MEZ!MEZ
18 MEZ!DET
19 MEZ!CO2
20 TAP!DET
21 TAP!CO2
22 TAP!Harvesting
23 DET!TAP
24 DET!Export
specifying the interactions among species or groups of species. These information
determine the network topology, which already provides some relevant insights
on the features of the ecosystem. It is normally represented as a directed graph
where each node represents a compartment and each arc denotes an interaction
between the source and target nodes. More precisely, an arc from node A to node
B represents a ow of energy or biomass from A to B. A common convention
is to depict dissipation for some node with an arc outgoing from the node and
ending in the ground symbol of electrical circuits [
The graph of a simple planktonic trophic network of the Venice Lagoon,
taken from [
This trophic network has some peculiarities: { dissipation (respiration) of PHP is not considered because the ow from CO2 to PHP models the net photosynthetic production, known from experimental data, i.e. the CO2 needed for respiration has been already subtracted; { ow from BPL to DET (mortality of BPL) is not considered because it is known to be negligible by experimental data; { ows from TAP, MEZ and MIZ to DET include both natural mortality and production of faeces; { ow from PHP to DET indicates only mortality, because PHP does not produce faeces; { in the case of MIZ and MEZ cannibalism is represented by arrows exiting and entering in the same compartment ( ows 12 and 17).
From the topology of the graph, or the corresponding adjacency matrix, some
information about system behaviour can be derived. Clearly the adjacency matrix
does not represent the information on weights of the interactions. For this reason
various other matrices have been de ned and used for analysis purposes, such as
the matrix of dietary coe cients, the Leontief structure matrix, the total
dependency matrix and many others which express di erent views of the network in
relation to structural and quantitative dependencies among compartments [
To move from purely topological analysis of a trophic network to quantitative
analysis, ecologists need quantitative data. Estimation of biomass and knowledge
of several rates (e.g. production rate, consumption rate, respiration rate, etc.)
are needed to quantify ows among compartments, together with quantitative
knowledge about diet composition of each living compartment. Some
information on primary production, speci c consumption rates and diet compositions
can be gained from eld and laboratory studies but it is unfeasible to
determine the magnitudes of all ows in the system directly. It becomes necessary,
therefore, to estimate the magnitudes of some of them by indirect means. A
helpful approach for estimating unknown ows consists in assuming the
balance of inputs and outputs for each compartment. If a su ciently long time
period is considered, mass balance in each node of the network is a reasonable
assumption because of the conservation of mass principle. Under the mass
balance assumption, the system is represented as a steady state snapshot of energy
ows, averaged over time. Di erent techniques are used for the trophic network
reconstruction, that is to infer unspeci ed ows by solving the balance
equations and satisfying the constraints among the ows in the system. The problem
is generally underdetermined and an in nite number of solutions comply with
the data set and the mass balance assumption. One technique is the Inverse
Model (IM), which has been rstly applied to trophic network in [
Several analyses on ecological networks have been de ned in the last decades.
Some of them are based only on the topology of the model, for instance
determining food chain length, connectance and the presence of cycles. In a balanced
model it is possible to study both qualitative and quantitative properties
measured by global system status indexes such as degree of recycling [
Petri nets are a well known formalism originally introduced in computer science
for modelling discrete concurrent systems. Petri nets have a sound theory and
many applications which are not limited to computer science (see, e.g., [
We denote a basic Petri net by N = (P; T; W; M0), where P = fp1; : : : ; png is the set of places, T = ft1; : : : ; tmg is the set of transitions, W : (P T ) [ (T P ) ! N is the weight function and M0 is the initial marking of the net, an n-dimensional integer vector assigning to each place its initial number of tokens.
We write t for the pre-condition of a transitions t, namely the n-dimensional vector t = (i1; : : : ; in), where ij = W (pj ; t) for j 2 f1; : : : ; ng. Sometimes it will be confused with its support, i.e., the set of places fpj j ij > 0g. The post-condition t+ = (o1; : : : ; on) is de ned dually.
The incidence matrix of a Petri net N , denoted by AN , is the n m matrix which has a row for each place and a column for each transition. The column associated with transition t is the vector (t+ t )T , which represents the marking change due to the ring of t.
Depending on the available information, Petri nets may permit to represent and study a system qualitatively, based only on the graph structure, as much as quantitatively or dynamically. An interesting structural analysis is based on the incidence matrix and it aims to determine the so-called invariants of the net. We focus here on T-invariants. Let N be a Petri net, with m transitions and n places, a T-invariant (transition invariant) of N is a multiset of transitions whose execution starting from a state will bring the system back to the same state, namely it is an m-dimensional vector in which each component represents the number of times that a transition should re to take the net from a state M back to M itself. It can be obtained as a solution of the equation AN
X = 0; where X = (x1; : : : ; xm)T
and xi 2 N; for i 2 f1; : : : ; mg:
A T-invariant X 6= 0 indicates that the system can cycle on a state M enabling
the cycle. As discussed in [
Two subclasses of Petri nets will be of interest in the modelling of trophic
networks [
Petri nets supply an executable speci cation: in the case of basic Petri nets,
we can play the token game, i. e. the non-deterministic ring of all the
enabled transitions. More sophisticated and realistic models and simulations can
be obtained through extended Petri net models. The most interesting in our
context are Continuous Petri nets. In Continuous Petri nets [
We start discussing how Petri nets can be used to model and analyse a trophic
network. We assume to know only the species (or compartments) and their
relations, which is the minimal knowledge generally available on a trophic network.
As a running example, we consider the trophic network TV of the Venice lagoon
in Figure 1. We illustrate how to build corresponding Petri net models and
discuss what we can obtain by applying some Petri net analysis techniques. We
use the tools Snoopy [
Given a trophic network T , a simple Petri net model can be immediately derived by replicating the topological structure of T in the Petri net. Recall that in the graph representation of T each species (or compartment) is a node and a relation between two species is a directed arc representing the ux between the two species.
A structural Petri net model of a trophic network T is the net Ns (T ) where { any species (or compartment) becomes a place; { any ow (relation) between two species S1 and S2 in T , becomes a transition having S1 as a pre-condition and S2 as a post-condition. { any outgoing ow from a species S1 to the external environment (e.g., dissipation) in T , becomes a transition with pre-condition S1 and empty postcondition; similarly, any incoming ow from the environment to a species S2, becomes a transition with empty pre-condition and post-condition S2. In absence of any information regarding the uxes, all weights are set to one. Transitions corresponding to interactions among species are referred to as internal transitions, while those corresponding to interactions with the environment are referred to as interface transitions. Note that the structural Petri net model of a trophic network is a free choice Petri net and, when restricted to internal transitions, it is a state machine Petri net.
By applying the described construction to the running example TV in Figure 1, we obtain a structural Petri net model which is depicted in Figure 2 (for the moment, please ignore the rates associated with transitions). The net includes six places (in yellow) representing the six compartments (DET, PHP, BPL, MIZ, MEZ, TAP) of the trophic network, and by as many transitions as the ows of biomass, to which we associate di erent colors to improve readability. More speci cally, respiration ows (producing CO2) are represented by light blue transitions; defecation ows are represented by brown transitions; mortality ows are represented by purple transitions; input and export ows for DET, as well as the harvesting ow for TAP are represented by red transitions; predationprey ows are represented by white transitions.
Note that transitions PHP CO2, representing respiration of PHP, and BPL DET, representing BPL mortality in the Petri net model of Figure 2, do not have a direct match in the trophic network TV of Figure 1. This is due to the fact that, as already mentioned, TV was simpli ed by taking into account also some experimental data. More precisely, the ow corresponding to PHP CO2 was integrated in CO2 PHP (modelling CO2 needed for photosynthesis) and BPL DET was considered irrelevant and thus omitted. 4.2
Structural analysis of trophic networks modelled as Petri nets Since the structural Petri net model strictly adheres to the graph representation used by ecologists, it obviously enables the usual structural analyses for trophic networks, for example to determine food chains lengths and connectance.
In addition, standard structural analyses for Petri nets can be used, like those based on T-invariants. The presence of T-invariants in a Petri net model of a trophic network is ecologically of interest as it can reveal the presence of steady states. The set of transitions involved in a T-invariant can be seen as a subsystem of the original system, whose equilibrium is autonomously maintained.
Given a trophic network T , consider the set of semi-positive T-invariants
of the structural Petri net model Ns (T ) and the corresponding Hilbert basis
B(Ns (T )), consisting of the minimal T-invariants. According to the terminology
in [
If we consider the elements of the basis, then for any such T-invariant I = (x1; : : : ; xm) we have xi 1 for all i 2 f1; : : : ; mg, namely each transition occurs at most once and the invariant is a set rather than a proper multiset. Moreover, since Ns (T ), when restricted to the internal transitions, is a state machine, for any pair of transitions ti; tj in the same invariant, whenever they share a place in the pre-condition or in the post-condition, they coincide. Therefore: { Minimal internal invariants are simple cycles, involving only internal transitions. { Minimal I/O invariants are acyclic paths, connecting two interface transitions.
In both cases we recover well-known concepts in trophic networks as presented,
e.g., in [
In our case study, the structural Petri net model has an Hilbert basis consisting of 69 minimal T-invariants, nine are internal and sixty are I/O invariants. The internal T-invariants are shown in Table 1. The rst two invariants describe the self-predation (cannibalism) of MEZ and MIZ. All the other T-invariants \traverse" the DET place, pointing out that Detritus is the way for recycling matter in this network. The I/O invariants start from source transitions CO2 PHP and input DET and end in sink transitions PHP CO2, BPL CO2, MIZ CO2, MEZ CO2, TAP CO2 and TAP harvesting. They model trophic chains allowing for respiration of the various compartments and for input and output of matter. 4.3
In this section we re ne the structural Petri net model of a trophic network, turning it into a continuous Petri net model. What we obtain closely resembles the representation of the trophic network usually adopted by ecologists, where Inv no. Transitions 1 MEZ MEZ 2 MIZ MIZ 3 DET TAP; TAP DET 4 DET BPL; BPL DET 5 DET BPL; BPL MEZ; MEZ DET 6 DET BPL; BPL MIZ; MIZ DET 7 DET BPL; BPL TAP; TAP DET 8 DET BPL; BPL MIZ; MIZ MEZ; MEZ DET 9 DET BPL; BPL MIZ; MIZ TAP; TAP DET the system is at a steady state and the input and output ows in all the compartments are balanced (the mass balance assumption). The choice of considering a continuous extension is motivated by the fact that we are modelling uxes of biomass which better correspond to continuous uxes.
The continuous Petri net model is still derived only from the network
topology by exploiting the minimal T-invariants in a way similar to what is done in
[
Remark 1. In the structural Petri net model of a trophic network Ns (T ) any place has typically at least one incoming and one outgoing transition, otherwise the place would unnaturally correspond to a compartment with monotonically increasing or decreasing content. Under this assumption, Ns (T ) is covered by T-invariants, namely each transition in the Petri net belongs to at least one minimal T-invariant. In fact, when we exclude interface transitions Ns (T ) is a state machine, hence for any transition, if we follow the predecessors and successors we will get back to the transition itself (internal T-invariant) or to an interface transition on both sides (I/O T-invariant).
In order to associate rates with the transitions, we assume that each subsystem corresponding to a minimal T-invariant 1. is active and 2. performs all its transitions once per time unit.
The assumption that all minimal subsystems of an ecosystem are active is quite reasonable from an ecological viewpoint. On the contrary the assumption that all subsystems perform all their transitions exactly once per time unit is rather strong and unrealistic. This is the simplest choice which can be taken in absence of further information on the ecosystem. When additional knowledge is available, it could be integrated in the model, as shown in the next section.
Let us consider the structural Petri net model Ns (T ) of a trophic network T as described in Section 4.1 and its Hilbert basis B(Ns (T )). According to the assumptions above, the rate of a transition t should depend on the number of minimal invariants in which t occurs. Then, for the trophic network T , we de ne the simple continuous Petri net model Nc(T ) as the continuous Petri net obtained by considering the structural model Ns (T ) as underlying Petri net and by associating to each transition t a constant rate given by:
rate(t) = jfIijIi 2 B(Ns (T )) ^ t 2 Iigj.
With such rates, all the transitions in all the invariants in Nc(T ) are performed once in one time unit and the system is in a steady state. Moreover, since all transition arcs are 1-weighted, rates and ows per time unit coincide. Remark 2. The continuous Petri net model of a trophic network satis es the mass balance assumption, namely, for all compartments the sum of ingoing and outgoing uxes coincide. This is an immediate consequence of the fact that minimal T-invariants are simple cycles or paths. Hence, given a place p, for any invariant Ii that \crosses" place p, one token is added to p by a transition in Ii and one token is consumed by another transition in Ii, namely the ux owing through p via Ii is balanced. This holds for any invariant crossing p and for any p. Therefore, the input and output uxes coincide for any place of the network.
In the simple continuous model Nc(T ), the system is represented in a steady state, with the uxes of biomass balanced in all compartments. This corresponds closely to the ecologists representation of a trophic network as a snapshot of the system at steady state. Note that the continuous Petri net model Nc(T ), despite the fact that it makes explicit some additional features, is still based only on the topology of T : biomasses do not play a role in the de nition of the rates.
For our case study, the continuous Petri net model resulting from the construction outlined above is shown in Figure 2, where each transition have an associated rate. Note that all places are balanced. We would like to validate our simple continuous model by considering some basic ecological processes and check their plausibility from an ecological point of view. For each compartment we compute the throughput, namely the total amount of ux owing per unit of time, in order to measure the degree of activity of the compartment. Besides we compute food consumption (total amount of ingested food per time unit), food assimilation (amount of ingested food minus amount of faeces, per time unit), respiration and mortality as percentages of the consumption. Table 2 shows the throughputs, the assimilation and respiration values as resulting from the model compared with those found in the literature.
The values derived from the simple continuous model are quite interesting. Considering the throughput, the various compartments are ordered as follows:
DET>PHP>BPL=TAP>MIZ>MEZ.
We may distinguish two main groups: lower trophic level compartments (DET, PHP and BPL), having higher throughput, and higher trophic level compartments (TAP, MIZ and MEZ), having lower throughput. This is coherent with the general knowledge on metabolic and growth rates of the two di erent groups of compartments under consideration.
Assimilation of the top compartment TAP is just over the maximum indicated in the literature, while assimilation requirements for MEZ and MIZ are perfectly met. However, MEZ assimilation is close to the lower bound of the indicated range. This is due to the fact that MEZ is a top level compartment in the network and no predators are modelled for it. This is a quite unrealistic assumption: in natural systems MEZ are actually preyed by other species, like shes. By adding an external predation on MEZ, we found that its assimilation becomes close to TAP and MIZ assimilation values.
Concerning respiration, TAP and MEZ satisfy the constraints found in the literature, while MIZ and BPL are slightly below the indicated value. Respiration of PHP is instead largely below the lower bound of the indicated range. The low respiration ows for MIZ, BPL and PHP is caused by the fact that there are only a few I/O minimal invariants involving these compartments. This is a misbehaviour of the simple continuous model, that must be somehow overcome.
Concerning mortality, for BPL it is irrelevant and this is in accordance with experimental data (see discussion in Section 2). Mortality of PHP is instead quite high: this is probably due to the fact that some PHP grazers, like shes usually occurring in lagoon systems, are not modelled.
On the whole, the continuous Petri net model realistically reproduces the
main processes of the trophic network considered in the case study. Even if it
based only on the network topology, it allows for deriving some quantitative
information on trophic network ows, which are coherent with results of
experimental measures taken in natural ecosystems. Moreover, the quantitative
validation shows that the model is somehow incomplete, signalling that two
further predation uxes, one for MEZ and one for PHP, should be represented in
the model.
MEZ
MIZ
BPL
PHP
DET
28
37
41
49
58
[
Model values Respiration = 36% Assimilation = 73% Defecation and Mortality = 27%
Respiration = 37% 80% Assimilation = 39%
Defecation and Mortality = 61%
Respiration = 14% 80% Assimilation = 78%
Defecation and Mortality = 22% Respiration = 17% Assimilation = Consumption
Mortality = 2,4% 30% Respiration = 2%
Assimilation = Consumption Mortality = 22% not relevant
Compartment throughput Literature values
TAP 41 [
[
We next examine more closely these two alternatives and apply them to our case study.
Speeding-up subsystems. Recall that any linear combination of minimal T-invariants is a T-invariant and a possible steady state of the network. Let us consider a generic linear combination of all minimal T-invariants:
X Ii2B(Nc(T )) kiIi ; ki 2 R:
The simple continuous model Nc(T ) corresponds to a steady state given by a linear combination of all the minimal T-invariants where all the ki are set to one. The re ned continuous Petri net model, Ncs (T ), is obtained from Nc(T ) by dropping the assumption that all subsystems have the same speed and setting the invariants constants ki to values possibly greater than one. In Ncs (T ) the rate associated with each transition is generalised to:
X rate(t) =
ki:
Ii2B(Ns (T )); t2Ii
The re ned continuous Petri net model Ncs (T ) still represents the trophic network at a steady state and with all compartments balanced, since the input and output uxes are balanced in each place for each minimal T-invariant.
We applied this idea to the case study and speed up the invariants involving respiration of PHP, BPL and MIZ, since the respiration ows of these compartments do not satisfy the ranges indicated in the literature (see Table 2). The new rates for the transitions are shown in Table 3.
Concerning PHP, it receives in input CO2 and partially release it for respiration. The unique I/O T-invariant for this process is fCO2 PHP; PHP CO2g. By speeding up this invariant to run ten times per unit of time, the respiration ow for PHP becomes the 16% of its total consumption, within the range indicated by the literature (see Table 2).
Concerning BPL, it is fed by the Detritus and part of the ingested food is used for respiration. The invariants involving transition BPL CO2 are fCO2 PHP; PHP DET; DET BPL; BPL CO2g and finput DET; DET BPL; BPL CO2g. By allowing the second invariant to run three times per unit of time, respiration of BPL become the 20% of its total consumption.
Concerning MIZ, we could speedup the invariants involving MIZ CO2, namely fCO2 PHP; PHP DET; DET MIZ; MIZ CO2g and finput DET; DET MIZ; BPL MIZg. By allowing them to run three and two times per unit of time, respectively, respiration of BPL becomes the 20% of its total consumption. Assimilation of BPL becomes the 80% of the consumption, still in the range indicated in Table 2.
Including constraints in the model. The second alternative for improving the model consists in \embedding" into the continuous Petri net model of the trophic network some available information regarding the metabolism of the species or their diet. We work under the simplifying assumption that ux constraints imposed on the model are linear. This assumption is generally satis ed by the constraints on metabolic uxes and on the diet partitions. For our case study, some metabolic constraints taken from the literature are given in Table 2.
We de ne a continuous Petri net model which structurally coincides with Ns (T ) and whose transition rates satisfy a set of linear inequalities. As in the previous cases, the transition rates are derived from the \speed" ki of each minimal invariant, but now we are interested only in invariants that satisfy the constraints. These can be obtained as solutions of a system of inequalities AN C X where AN is the incidence matrix of Ns (T ). We can consider the minimal such T-invariants, referred to as the constrained Hilbert basis BC (Ns (T )), so that any solution of (1) will be a linear combination of elements in BC (Ns (T )).
A continuous Petri net model Nc(T ; C) for the trophic network T satisfying the constraints C is de ned as follows. The underlying Petri net is Ns (T ) and each transition t is associated with a constant rate:
rate(t) = jfIi : Ii 2 BC (Ns (T )) ^ t 2 Iigj: In this way each transition in each constrained invariant Ii in BC (Ns (T )) can be performed once in one time unit.
When applied to our case study, this approach produces a linear system of equalities and inequalities, where the inequalities express the literature knowledge summarised in Table 2. By considering only the inequalities given by the lower bounds, the constrained Hilbert basis contains 349 minimal invariants. The induced rate constants for the extended network automatically satisfy the given ecological constraints.
The two approaches could be combined, by determining the constrained invariants and by setting for them possibly di erent speeds.
Simulations on continuous models with constant rates do not provide meaningful information. Some hints on how to further re ne the model to do simulation analyses are given in the conclusions. 5
In this paper we explored the use of Petri nets for representing and analysing trophic networks and our preliminary results are encouraging. A trophic network naturally translates into a structural Petri net model which allows for recovering classical trophic networks concepts and analyses. The structural model can be re ned into a continuous Petri net model that closely resembles the representation of the trophic network usually adopted by ecologists, where the system is at a steady state and the input and output ows are balanced in all the compartments. Despite the fact that the Petri net models proposed are still simplistic (in particular, the continuous models have constant rates, independent of the masses), in our case study of the Venice lagoon, the analysis of the continuous Petri net model shows that it realistically reproduces the main ecological processes. Furthermore, it shows that the continuous Petri net model can be fruitfully used for an early stage validation of the trophic network under study. Two re nements of the continuous Petri net are considered: the rst is based on a ne tuning of the speed of the minimal T-invariants, while the second one is based on a systematic embedding of some ecological knowledge expressed as linear inequalities into the calculation of the Hilbert basis. This however might have scalability problems, since the constraints increase the size of the Hilbert basis, and the problem of determining the Hilbert basis is already in EXPSPACE.
Future work deals with making the Petri net model more realistic and dynamic, by adding biomass information on compartments. The knowledge of biomasses at a steady state can, in fact, be used to derive constants for a continuous model governed, e.g., by the mass action equation. We believe that introducing rates dependent on biomasses could allow for interesting simulations, describing, not only the steady state but also the transient behaviour leading to such state. Additionally, on such model perturbations of the biomasses and of the speed of the various interactions could be used for performing what-if analyses.
Acknowledgements. We are grateful to Monika Heiner and Andrea Marin for many inspiring discussions.