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8DA2AZDQV7ZE > Doc # Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays. Analysis of Deterministic Cyclic Gene Regulatory.

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As the resource e. We will therefore be concerned with the possibility of steady states, 42 2. We will start with a simple situation where the living system is characterized by a single descriptor, the biomass B. This growth rate depends on B only. There is positive growth at all positive levels of B, up to the carrying capacity, beyond which G is always negative. This would be a frontier situation, characteristic of, for example, an invasive species in its early history.

Higher B leads to decline in G. At carrying capacity, B is stable. Logistic Growth Figure 2. The steepest growth occurs midway in the trajectory, at intermediate B. Biomass 1 0. Time evolution of B and growth rate G in the absence of harvesting. If initial conditions are small, then A is big.

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Growth and harvesting rates. This is the maximum sustainable yield MSY. We will lump all these factors into a single effort variable E. The harvest depends on the effort and the biomass. Increases in effort, biomass, or technology increase the harvest, linearly in this case. Absence of any of these factors guarantees zero harvest.

If effort were able to be controlled, we could choose among these equilibria or any other E, B pairs along the line represented by Equation 2. Figures 2. The logistic growth function is compensatory at low B; the unstable intersection is absent. This growth curve is depensatory at low B, with an unstable descent to extinction in that low B range. We will take the sales price for the harvest, p, to be constant and the wage or cost of effort, c, also as constant.

Point K is also an exclusively biological one, with no harvesting effort and no rent. This resource is on the margin of a large economy, which it does not affect; p and c are constants.

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Aside: It is interesting to reexpress rent in terms of B and H. Which equilibrium is likely to occur? The opposite is easy to envision — for example, when c is very large and rent is always negative.

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A steady harvest does not pay. The condition of vanishing steady rent Equation 2. Effort may not increase without encountering negative rent, and vice versa. It is visually apparent that more rent may be earned by reducing effort relative to the open-system equilibrium B0.

Taking the derivative of Equation 2. A standard characterization is that there is less work being done, more money made, and a higher biomass.

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The effect on harvest H is ambiguous and depends on the parameters. Table 2. Exploitation during this period is lucrative. Figure 2. The equilibrium solutions are unchanged, yet all hope for monotone solutions is gone in these rapid-response-torent scenarios. A complex periodicity of about two years is apparent.

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Continuation of Figure 2. Program Fish1. These are autocorrelated as in the Appendix. The harvest at B is then sustained, and its sale value is pG B. Otherwise, the sustainable solution is between extinction and the MSY point. It is clear that the scenario given initially can be relaxed; starting from any initial B, we arrive by the same reasoning at the desired equilibrium: balancing the annual yield of the initial harvest against the annual yeild of steady harvesting.

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Notice that here we have no explanation for r, unlike the biology that creates G. A fuller examination of this balance would need to develop the relationship between the growth rate of money invested and the existence and growth of natural resources. These are dramatically different situations! Illustration of an instantaneous surge in E and the resulting change in B.

Both levels of B are to be harvested sustainably. On an annual basis, the net 2. This result generalizes that.

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This is the intemporal, rentmaximizing equilibrium for the base case logistic fishery. Programs Fish1. In this formulation, the cost of harvesting increases as the biomass is reduced. It recovers the simple costless case, Equation 2. Equation 2. A realistic treatment of technology is needed for a full theory of natural resource dynamics; here we simply speculate on the form it might take. Recall this is a rate of innovation. Candidate innovation rate functions. The abrupt termination there would be better modeled with a gentler approach to the origin. The technology grows during exploitation.

But ultimately, we must confront its effect on the growing resource B. The dynamics of this system are illustrated in Figure 2. Increasing technology h moves the equilibrium toward the left lower B unless compensated for by decreasing effort E. Dynamic adjustment as in Figure 2.

In this form, S is positive for all positive values of h, decreasing monotonically. These works and many other typically go much further into the description of biological populations, resonating well with the subsequent Chapters 3 and 4 herein. Critical issues reveal the necessity of sustaining the ecosystem; the biological populations hosted; and the economic interactions with people. Achieving this set of outcomes will require a high degree of practicality in diverse cases. The simple model used here illumines the multiplicity of criteria that might be relevant: the biomass, the harvest, the rent, the jobs.

All have found their their place in practical systems. The critical issue of management of common-pool resources is fundamental to the considerations introduced here. The reader is referred to the important contributions of Ostrom et al. These are important contributions to natural resource management, touching fundamental social science issues broadly. Both time series are autocorrelated. Explain your results. Be careful about G here; it is not hard, just different.