Chapter 4 Probabilistic Nature of Individual Behavior
The intricacies of a biological system emerge at the individual level. What properties should simulations possess to capture the structure and dynamics of a population of individuals? While a biological system has a complete state description across all individuals at any instant, individuals only have partial knowledge and must make probabilistic choices about future behavior. Each individual is partially oblivious to the `optimal’ sequence favored by the biological system to improve its quality of life.
Each individual has a biased view of its environment, and operates
with information that is subjective, incomplete and at times
erroneous. The system is stochastic or fuzzy' (Zadeh 1973) since future events for an individual are subject to random decisions based on imperfect system knowledge. Each individual has instincts, or strategies, that evolved as species characteristics. In addition, it possesses certain morphological and ethological features based on its unique heredity that may improve species, but not necessarily individual, survival. Someprimitive’ entities can have reasonably
constant instincts during their lifespan. Highly developed individuals
experience periods of learning that transform their ability to exploit
the environment.
Holling (1966) analyzed the functional response of the praying mantis (Hierodula crossa, Giglio-Tos.) to a population of houseflies (Musca domestica) finding that hunting behavior is intrinsically probabilistic. Holling decomposed mantid predation into three basic components:
Rate of successful search, depending on (a) reactive distance of the predator to prey, (b) proportion of prey successfully attacked, and movement speed of the (c) predator and (d) prey.
Time prey are exposed to predator, depending on activities (a) related and (b) not related to prey feeding.
Time spent handling each prey, including (a) pursuing and subduing, (b) eating, and (c) digesting afterward.
Holling’s analysis suggests that predator-prey interactions are sequences of events involving individuals, not densities. They occur only under certain conditions that depend on the morphological and ethological characteristics of both predator and prey. A predator-prey interaction event depends on many conditional probabilities that in turn depend upon the states of a particular pair of predator and prey.
Since an interaction is inherently probabilistic, uniquely dependent upon the individuals involved, how can we transform these discontinuous, heterogeneous, probabilistic data into piecewise continuous predator-prey models? Analytical solutions surely fail. Simulations tend to judiciously ignore as many variables as possible to reduce dimensionality. This introduces uncertainty by sampling the state space, making the system more probabilistic.
The usefulness of any model depends on its ability to suggest clearer understanding of the data collected. Conversely, modeling of a biological system is intrinsically dependent on what is measured and how individuals are sampled. Sparse sampling `quantizes’ a biological system, restricting what facets of an individual are studied, leading to incomplete descriptions of the dynamics of each organism. The researcher observes certain individuals and assigns them to a sequence of quantifiable states. Unobserved individuals have unknown states, leading to a probabilistic projection of the whole population onto the states of those that are observed. Thus the measurement process chosen by the biologist limits the class of models for the structure and dynamics of the biological system. Further, the process of measurement and understanding of that system evolves over weeks or years of investigation.
4.1 Properties of an individual
Since individuals are paramount to our approach, we explore more thoroughly what an individual is in this section. An individual is so complex that any attempt to simulate that organism and its dynamics is extremely difficult. Modeling the dynamics of a population at the individual level requires defining properties of an individual at an arbitrary operational level. The following twelve properties define an individual as a higher level living system. They are by no means complete, but they limit the type of modeling effort that would be acceptable:
- An organism is inexhaustibly complex.
- Its state cannot be completely determined.
- It has no natural state-space representation.
- It is unique relative to all other organisms.
- It changes continuously and irreversibly.
- It is unique relative to its own entire past and future.
- It is highly organized internally.
- It is sharply differentiated from its environment.
- It exchanges mass, energy, and information with its environment.
- Its perception of its environment is incomplete, abstract, and specialized.
- It has memory of its past that modifies its present behavior.
- It responds to its environment as a discrete integrated unit.
Properties 1, 2, 3, and 7 are related. The internal system of an individual is operationally indefinable. A particular organism can be partially described in terms of its constituent parts, for instance its nervous system. The dynamics of neural systems helps unravel the mechanism for transmitting information, but may not be sufficient to study memory, which further involves the dynamics of RNA. Under normal conditions an organism receives stimuli from hundreds of sources, processing based on its unique memory and physiological capacity in a nonlinear, non-equilibrium and non-decomposable fashion. A `complete’ understanding of these internal dynamics would involve a hopelessly nonlinear system with hierarchical structure.
Properties 4, 6, 8, 11, and 12 claim that organisms function as unique, complete individuals. Each behaves as a unit under a set of stimuli able to react uniquely depending upon its present state and partial memory of past events. Properties 5 and 9 imply that an organism converts mass, energy and information, obeying entropy, but is never at equilibrium until it dies. Property 10 says that each organism has finite memory, cannot recall all sensed stimuli, and can respond to only a few available stimuli.
4.6 Heisenberg uncertainty principle
We have approached the concept of a `measurement’ from a mathematical point of view, and not from the point of view of a field biologist. The quality of what is called measurement is similar to the concept of a decomposition of a Hilbert space in which the measurement process and the observer is in some sense an active contributor to the determination of the understanding of the system being studied. We are not proposing the existence of some sort of Heisenberg uncertainty principle of canonical variables. Rather we suggest that knowledge of the structure and dynamics of a particular system is uniquely determined by the measurement that a field biologist performs. Further the process of measurement and understanding of that system is itself a dynamic process. For the field biologist, the usefulness of any modeling technique is directly related to its ability to suggest clearer understanding of the data collected.
4.7 Games of Skill and Chance
Games imitates life. How do the properties of common games reflect aspects of a biological system that might be relevant for simulation modelling? In the development of Ewing’s approach certain properties of the biological system–when viewed at the level of resolution of the individual–became apparent. These properties are examined in the next three subsections.
4.7.1 Backgammon
Backgammon is an extremely old game which has been played for centuries. The game has three interesting properties. First, at any given time t, there exists a complete description of the state of the game. Second, all moves are inherently probabilistic, that is, all moves are made by the toss of a pair of dice. Third, given an acceptable structure, there exist certain strategies which will minimize the chances of `success’. Certain strategies are preferred because the randomizing process is repeated a large number of times over the acceptable structure. The game itself is a true game of chance since at any point in the game, the next step is essentially random.
The biological situation is similar if we resolve a given population to the level of the individuals. Ewing et al. (1974) asserted that from this level of resolution any individual is characterized by twelve operational properties, listed below:
From these properties, it is apparent that any individual is so complex that, at any given instant in time, the behavior of the individual is probabilistic relative to any subset of events which are sampled. Furthermore, the probabilities at a given instant will change as a function of the events which occur to that individual. The probabilities themselves have a system of probabilistic feedback loops.
At any one instant, the randomizing process essentially forces a quasi-equal probability onto the system; that is - complete chaos. However, over a large number of events, the structure inherent in the biological system become less opaque. It is this structure that favors certain strategies. An individual’s ability to apply these strategies to his situation in the system determines his ultimate success or failure in the system.
4.7.2 Bridge
Bridge is the second game whose properties we wish to examine. Unlike backgammon, bridge has a randomizing process which is performed once. At the beginning of each hand, a certain participant randomizes (shuffles) a deck containing fifty-two cards and the cards are dealt to each participant. A complex sequence of bidding occurs in which part of the structure of each of the participant’s cards are revealed. At some point in the game the bidding stops and an exchange of cards takes place. With each exchange, the structure of each of the players hand becomes more apparent, and each player attempts to exploit his understanding of the particular structure inherent in the game. Each player is, in essence, operating with incomplete information. Further, it’s his understanding of that information which determines his tactics–not the actual information. In the analogous biological situation, each individual operates with information which is subjective and incomplete. An individual has revealed to it, at any given moment in time, only a pat of the environment in which it exists. That individual is operating on the basis of incomplete and, at times, erroneous information – i.e., the information is `fuzzy’ (Zedah, 1973). In a realistic biological system, an individual has a biased view of his environment. Through the individual’s evolutionary processes, it possesses certain instincts which are characteristic of its species and through its unique hereditary chain it possesses certain morphological and ethological characteristics. In extremely primitive entities the initial instincts may be constant for its total lifespan. For more highly developed individuals, long periods of learning can take place in which the individual’s ability to exploit its environment changes as its awareness of the structure of its environment changes, and as the actual environment changes.
As in backgammon, the individual must exploit its view of its local environment. The researcher must be aware of the individuals incomplete understanding of its environment, but that his own subjective understanding of the biological system is also incomplete. In some senses the researcher is an observer to the game in which he is not allowed access to all the rules nor can he look completely at any one player’s hand, yet he must predict the outcome of the game. He has, therefore, an intrinsically different understanding of the system. His view of the system determines his measurement of that system (how does he choose to look at a given participant’s cards). The measurement process is intimately connected to the researcher’s view of the biological system in question and together they determine his predictions. The researcher is again faced with a probabilistic view of a given population which is resolved to the level of description of the individual. The probabilistic approach is necessary to handle the problem of incomplete information.
4.7.3 Chess
Chess is a game of skill played by two players. Chess is, in
principle, deterministic. There exists a full set of rules with
which both players are familiar. At any instant, the complete state
of the game is, in principle, available to both players, and that
state space is finite.
Why then, is chess considered a complex game? If one wished to
examine each possible move for each player for the whole game, i.e.,
determines each trajectory through the state space for the whole game;
one is faced with an enormous decision tree. For a complete game
there is on the order of 10100 to 10200 possible paths through the
state space. Therefore, even though the game of chess is, in
principle, deterministic and finite; it is, from a practical or
operational standpoint, inexhaustively complex. Because of the
inherent complexity, it is highly unlikely that any direct numerical
analog of the game is possible. It is also obvious that analytic
solutions are impossible. By necessity, the game is played
heuristically. Due to the high dimensionality of chess, any statement
of how the game progresses, is operationally a probabilistic
statement.
Suppose that is possible to define a biological situation – at the level of description of the individual – which is finite and deterministic. Ewing (Ewing, et al., 1974) has shown that even for descriptions with an extremely small number of individuals, the possibility of numerical solutions becomes operationally impossible rather rapidly. In such situations, there exists the time-honored technique of judiciously ignoring as many variables as is possible. By ignoring a certain subset of variables, a deterministic system becomes a probabilistic system. Because of the problem of high dimensionality in population biology, it becomes necessary to subsample the biological state space and the description of that population is necessarily probabilistic.