NetworkFlowMath

Much of the (managerial) pictorial language used in organizations can be expressed in differential form:

dN/dt = f (N)

This expression assumes that growth is continuous and that the growth rate dN/dt will be some function of a density N. The density N can be used to represent quantities like capital, resources, information, biomass, etc.

Thinking of any company as a population (of humans) organized somehow to survive together through being part of this organism (organization) a part of their time, we can use mass dimensions and use the form dN/dt = f (N) to describe the behavior of such populations. Biomass dimensions translate easily to population characteristics. Population can be all of the people employed by an organization or people in specific stakeholder populations. Growth of biomass in an organization can depend on a wide variety of interrelated factors. Some are non-human characteristics of the environment; some are characteristics from the culture and some stem from contact with other cultures.

For now I'm assuming all non-human factors remain constant -no economical waves or random environmental fluctuations- and will ignore age and spatial structures and behavioral variation: I'm considering an organizational population homogenous in every respect. For this population type we can write the growth equation in the form:

dN/dt = N [i (N) - e (N)]

Where i is the per capita immigration rate and e the per capita emigration rate of employees. One possible form for such curves is the following:

Where growth in the immediate future is determined by the value of (N). The per capita immigration rate inclines steeply from zero at low densities perhaps because of a capital injection when the company was formed. At a later stage in a companies' life and coming from higher densities its rapid decline at low densities can be the company caught in a death spiral. It has found no way to renew or change strategy effectively and can have chosen for a hiring stop.

The per capita emigration rate also increases rapidly at low densities, perhaps because of high stress levels when starting a company. At a later stage of the companies' life and coming from higher densities the steep decline can be related to downsizing tactics while being caught in a death spiral. With higher densities the per capita emigration rate increases gradually perhaps due to lack of resources or quiet places to think. The per capita emigration rate declines again after this second peak. This could happen, for instance, if group defense strategies with similar companies like non-competition treaties are used.

The resulting growth rate f = N (i-e) leads to the following system dynamics for such homogenously populated companies.

The dashed arrow indicates the direction of growth in the immediate future, in case the initial value of N falls in the region occupied by the arrow. For each value N of the population, the growth in the immediate future is determined by the value of (N).

In those regions of the N-axis where f (N) is negative, the dynamics dN/dt = f (N) will cause the population density to decrease. In those regions of the N-axis where f (N) is positive, the dynamics dN/dt = f (N) causes the population density to increase. If (N) = 0 for some population density N, then the value K is called an equilibrium value. If N has initially the value K, then it will remain at this value [since dN/dt =f (N,) = 0] until it is displaced away from this value by some external pressure. Equilibrium is almost always a point at which the curve f (N) crosses the N-axis. The slope of the curve f (N) is negative at equilibrium K:

df/dN|k < 0

If the population density is moved below K, it will be in a region of positive growth and increase back to K. If it is moved above K, it will be in a region of negative growth and decrease back to K. The tendency of the population's intrinsic dynamics is to push the density toward K once it gets close enough. Such an equilibrium K is called stable. Stability can be used in many different ways, and different concepts may be appropriate in different contexts. For example, the word "stable" can mean population composition (that the same employees are present each year) or the population density of a specific type of company remains constant. Or perhaps some "structural stability" is meant. In all of the above, "stable" is a property that is either possessed or not. It can have also been used in a relative sense instead, for example, calling a population whose density has a smaller coefficient of variation over time a "more stable" population.

In this case "stable" means asymptotically stable in the sense of Liapunov or neighborhood stable or locally stable. The set of population densities E < N < B is called the domain of attraction or basin of attraction of the equilibrium. At equilibrium E the slope of the curve f(N) is positive:

df/dN|e < 0

In this case, the population density will move away from its equilibrium value after the slightest initial displacement away from its equilibrium E, and the equilibrium is called unstable. E can be called an extinction threshold: if for some reason (bad economical climate, downsizing, whatever) the density falls below E, then the company will die and suddenly we're all without a job.

The slope of the curve f(N) is also positive at equilibrium B. This equilibrium is also unstable and can be called a breakpoint, for once we get past equilibrium B we're heading for the next stable state in the life of our case company.

So even with such simple homogenously populated companies we can have multiple domains of attraction. When a company and its environment are stratified and diversified I would expect multiple domains to be all the more likely.

-- NynkeEtkFokma, 2002