Hirooka has gathered some evidence for this time course, the most extensive being for the electronics industry. The application of this equation to innovation implies, perhaps a little surprisingly, that innovation grows autonomously that is, it does not need any adjunct (although, as written, it cannot start from zero-we may assume that it begins spontaneously with a lone innovator). This decomposition of the model variables illustrates the important point that the system components N, D, O, and E are not necessarily disjoint here we see that the variables x and y belong naturally to both components N and O. Finally, the environmental carrying capacities K and L comprise the external environment E. The decision-making institution D clearly is composed of the harvesting functions h 1 and h 2, whereas the observer O may be thought of as the variables x and y, since it is reasonable to suppose that the fish populations can be measured directly. ) are harvesting functions.Ī plausible separation of this model into the macrocomponents outlined earlier is to consider the variables x and y, together with their growth rates r and s and interference parameters α and β, as the natural system N.Where x and y are the two fish populations, r and s are growth rates, K and L are maximum population levels the environment can support, α and β are measures of the extent to which each species interferes with the other's use of the external resource (food supply), and h 1( Next we explore a control algorithm that uses chaos to ensure favorable robotic behavior.ĭx / dt = rx ( 1 − ( x / K ) ) + α x y − h 1 ( t ), dy / dt = sy ( 1 − ( y / L ) ) + β x y − h 2 ( t ) , How many engineers have spent endless hours in lab running and rerunning experiments thinking something was wrong with the design or the experimental setup when in fact the bizarre behavior was inherent in the system itself?Īs important as it is to understand that chaos might manifest itself as undesirable behavior, it is also possible to harness its power for system design.
The same system, governed by the same modeling equation, displays three distinct behaviors: asymptotic stability, limit cycling, and chaos.Ĭhaos can have serious implications in engineering systems, and it is important to know this behavior exists. The logistic equation is quite interesting. In fact, making the initial condition differ by 0.0000000001 results in divergent trajectories it just takes a larger number of time intervals to start seeing the difference.
However, if the two initial conditions were started even closer to each other, say 0.2 and 0.201, the trajectories would still behave quite differently. One could argue that 0.2 and 0.3 are not very close. This is an example of sensitive dependence on initial conditions. Notice that the two trajectories are not similar at all. But that isn’t the only unusual behavior. In this case, the trajectory doesn’t settle to any observable pattern, even when the values are plotted for n up to 100.
Two trajectories are plotted with different initial conditions: x = 0.2 (solid line) and x = 0.3 (dashed line).įor r = 3.9, the system exhibits a completely different behavior. The emergence of chaotic behavior in the logistic equation as growth rates increase. When laypersons speak of overpopulation, often they are referring to exceeding the carrying capacity of the Earth.Figure 4.24. Most typically this is a response to stable enviro… Overpopulation, Overpopulation Arithmetic growth takes place when… K-selection, K-selection The selection for maximizing competitive ability, the strategy of equilibrium species. S-shaped Growth Curve, S-shaped growth curve(sigmoid growth curve) A pattern of growth in which, in a new environment, the population density of an organism increases slowl… J-shaped Growth Curve, J-shaped growth curve A curve on a graph that records the situation in which, in a new environment, the population density of an organism increases r… Exponential Growth, The distinction between arithmetic and exponential growth is crucial to an understanding of the nature of growth.