# Complexity, collective effects, and modeling of ecosystems

Does the concept of “Complexity” bear any specific meaning or is it just synonymous with complicated and yet not comprehended phenomena? We argue that it is possible and useful to use the term in a specific and reasonably well-defined way. It is useful because a number of common trends and implications become clear when a phenomenon is classified as part of “Complexity Science.”

The science of complexity emphasises the interactions between components. It stresses that components, most often, are heterogeneous and evolve in time. Complexity is concerned with the emergent properties at systems level originating from the underlying multitude of microscopic interactions.

In an attempt to make our discussion more cleawe will immediately describe the way we use some terms central to our exposition. We hurry to stress that these descriptions are not meant to be exhaustive final philosophical definitions, but rather intended to lower the risk of misunderstanding when we deal with terms frequently used to mean differentthings by different people. And now our specifications.

Complex Systems consist of a large number of interacting components. The interactions give rise to emergent hierarchical structures.

The components of the system and properties at systems level typically change with time. A complex system is inherently open and its boundaries often a matter of convention.

Statistical Mechanics seeks to understand how properties at systems level emerge from the level of the system-components and their interactions. This often involves the application of probability theory, and a number of mathematical techniques. Throughout, we draw a distinction between statistical mechanics and statistical physics. The latter is mainly concerned with the microscopic foundation of thermodynamics and, for example, phenomena such as phase transitions and superconductivity. We consider here statistical mechanics as a mathematical methodology, which can be applied to many different sciences including economics, population biology and sociology, to name a few.

In the next section, we will for concreteness illustrate our arguments by briefly describing a complexity inspired model of evolutionary ecology called the Tangled Nature model. This will allow us to demonstrate how macroevolution can be modeled as emerging from the interacting microevolution, which consists of individual organisms influencing each other and undergoing reproduction which is prone to mutation. We will discuss feedback, emergence, network structures, and the intermittent temporal mode of macroevolution in contrast to the steady smooth pace of dynamics at the level of individuals.

## Tangled Nature Model

The Tangled Nature model is defined at the level of interacting individuals. It is an attempt to identify possible simple mechanisms behind the myriad of complicated interactions, feedback loops, contingencies, etc., as one moves from the short time reproductive dynamics at the level of individuals, to the long time systems level behavior. The strategy is to keep the model sufficiently simple to enable analysis, and to pinpoint the details or assumptions in the model that are responsible for the specific behavior at the systems level.

One major concern of the model has been to understand how the smooth continuous pace of the reproductive dynamics at the level of individuals, can lead to intermittent or punctuated dynamics at the level of high taxonomic structures. To be able to address such issues, the model considers individuals as represented by a single sequence with individual number $latex \alpha$, denoted by $latex S^{\alpha} =(S^{\alpha}_1,S^{\alpha}_2,\ldots,S^{\alpha}_L)$ belonging to a sequence space $latex S$, where all $latex S^{\alpha}_i \pm 1$. These sequences undergo simple reproduction during which a given sequence duplicates itself, and while this happens, components of the sequence may mutate, represented by the offspring having a different sign from the mother, that is, $latex S^{\gamma}_i=-S^{\alpha}_i$, where denotes the index for the daughter, and the one for the mother. A species will be identifies as a local peak in the density $latex n(S,t)$ The aim of the model is to understand the macrodynamics emerging at the systems level. This is done by analyzing the dynamics of the occupancy in this sequence or type space.

The system consists of $latex N(t)$ individuals, and a time step consists of one annihilation attempt followed by one reproduction attempt. A reproduction event is successful with varying probability $latex p_{off}$ , defined later, and an annihilation attempt is successful with constant probability $latex p_{kill}$ . The killing probability is considered a constant independent of type for simplicity. The individuality of the specific types, or sequences, is given by their ability to reproduce.

Because we are interested in the collective, or complexity, aspects of evolution, the Tangled Nature model stresses the mutual influence among different types of organisms. This is done by assuming that each individual of type $latex S$ is able to reproduce, when selected for reproduction, with a probability $latex p_{off} (S, t)$ that depends on the sequence $latex S$ and the configuration of other types in the type space. The reproduction probability, $latex p_{off}(S,t)$ , is determined by a weight function $latex H(S^{\alpha}, t) = \frac{k}{N(t)}(\sum_{S\in\mathcal{S}}J (S^{\alpha}, S)n(S, t))-\mu N(t)$, where $latex k$ controls the strength of the interaction (large $latex k$ means a large interaction), $latex N(t)$ is the total number of individuals at time $latex t$. $latex n(S,t)$ is the number of individuals (or occupancy) at position $latex S$.

Two positions $latex S_a$ and $latex S_b$ in genome space are coupled with fixed but random strength $latex J(S_a,S_b)$ which can be either positive, negative, or zero. This link exists (in both directions) with probability , that is, is simply the probability that any two sites are interacting. If the link exists, then $latex J(S_a,S_b)$ and $latex J(S_b,S_a)$ are both generated randomly and independently, and such that they belong to $latex (-1,1)$. To study the effects of interactions between species, we exclude self-interaction so that $latex J(S_a,S_a)=0$.

The conditions of the physical environment are simplistically described by the term $latex \mu N(t)$, where $latex \mu$ determines the average sustainable total population size, that is, the carrying capacity of the environment. This is an example of how the question of the openness and “surroundings” of ecosystems arises in a natural way in the present statistical mechanics like formalism. An increase in corresponds to harsher physical conditions.

We allow for mutations with probability $latex p_{mut}$ per gene we perform a change of sign $latex S^{\alpha}_i \rightarrow -S^{\alpha}_i$ during reproduction. Successful reproduction occurs with a probability per unit time, $latex p_{off}(S,t) \in [0, 1)$, given by $latex p_{off}(S,t)=\frac{\exp^{H(S,t)}}{1+\exp^{H(S,t)}}$.

After a short transient period the initial state becomes irrelevant. Natural selection will ensure that only certain configurations of occupied sites are viable. These are configurations for which the mutual interactions between the types lead to offspring probabilities that, for a significant part of the occupied types, are able to balance the killing probabilities, that is, $latex p_{off}(S, t)=p_{kill}$ for some set of types $latex S$.The dynamics in type space is characterized by a two-phase switching, consisting of long periods of relatively stable configurations (quasi-Evolutionary Stable Strategies or q-ESSs) interrupted by brief spells of reorganization of occupancy called transitions. Transition periods are terminated when a new q-ESS is found. The intermittent macrodynamics is not in a stationary state.

When one considers very many realizations of the dynamics it turns out that the transition rate between q-ESS decreases with the age of the system. This happens because selection is able to pick out configurations in type space that tend to possess more beneficial links than is the case between a randomly selected set of types. We consider this directedness of the long time systems level dynamics to be prototypical of complex systems.

Intermittent evolution of the occupancy in type space.Time is along the x-axis. The different types are labeled up along the y-axis. Whenever a type is occupied a dot is placed at its label. Long stretches of parallel lines indicate epochs during which the main composition in type space remains essentially the same.