The Food Chain Continued

Theorem 4.2. Let n — of the Floquet multipliers of (4.6) lie inside the unit circle in the complex plane. Then 7 is an asymptotically orbitally stable trajectory of (4.5). 

A trajectory 7 is unstable if one of the multipliers is outside the unit circle. Hyperbolicity, in this case, is taken to mean that exactly one multiplier is on the unit circle. When a periodic solution of (4.5) exists, one multiplier of the linearization is always equal to 1. This is where (4.4) is useful, particularly when the system is two-dimensional. 

We return now to the discussion of Section 3. A criterion for the stability or instability of the rest point had been obtained in the form (3.4). If the rest point is locally asymptotically stable, it is possible that there could still be limit cycles surrounding it. The following arguments show that this cannot happen. These arguments are very detailed and very tedious the reader who is not interested in the technique might be well advised to skip to the statement of the main theorem. 

In the proofs that follow, the quantity 1 — a (/)— y(/) turns up so frequently that we use the equivalent (in terms of the original system) S(t) to keep the computations simpler. Similarly, we use = 1 -x -yc in order to shorten the notation. We begin with the following computation. 

To avoid excessive mathematical notation the next lemma is stated in terms of the system 

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Betalain colorants are well established in the food chain and will probably continue in a limited capacity. 

It turns out that most of these compounds have similar characteristics that contribute to their toxicity to both humans and other species of plants and animals. First, the compounds are environmentally persistent. Many of the early pesticides, and certainly the metals, do not break down in the environment or do so only very slowly. If persistent chemicals are released continually to the environment, the levels tend to rise ever higher. This means they are available to cause harm to other organisms, often not even the target of the pesticide. Second, the early pesticides were broad acting and toxic to many species, not just the target species. These poisons often killed beneficial insects or plants. Third, many of these compounds would bioaccumulate or concentrate in species as they moved up the food chain. The chlorinated pesticides accumulate in the fat of animals. Animals that consumed other animals accumulated more and more of these pesticides. Most species could not metabolize or break down the compounds. Lead accumulates in bone and methyl mercury in muscle. And finally, because of their persistence in the environment and accumulation in various species, the persistent toxicants spread around the world even to places that never used them. Animals at the top of the food chain, such as polar bears and beluga whales, routinely have fat PCB levels greater that 6 ppm. 

Proteins are nature s polyamide condensation polymers. A protein is formed by polymerization of o-artiino acids, with the amino group on the carbon atom next to the carboxylic acid. Biologists call the bond formed a peptide rather than an amide. In the food chain these amino acids are continuously hydrolyzed and polymerized back into polymers, which the host can use in its tissues. These polymerization and depolymerization reactions in biological systems are all controlled by enzyme catalysts that produce extreme selectivity to the desired proteins. 

Statutory legislation to control the levels of such substances in food has been introduced in the UK and elsewhere. In more recent years, other potentially toxic elements have come into focus. Lead, cadmium and mercury have been the subject of much monitoring of the food chain and other metals, in particular aluminium, are continuing to attract attention. Nitrate and nitrite in food from food additive use is regulated across the European Union, but its presence in food crops has raised concerns. 

POPs in the environment can enter the food chain, bio-accumulate and bio-magnify as they move up the trophic levels and ultimately end up in the human body. It is expected that POPs will continue to accumulate in the body fat and their average concentrations will increase with age. The level of POPs contamination in human blood/serum and breast milk can serve as a good indicator of their body burden. 

In many countries, both programmes measure the environmental radiation field and radioactivity in air, water and the various elements of the food chain the objective is to enable continuous assessments to be made of the impact of radioactivity on health, and on the environment. In some countries, the two programmes are complementary so that country-wide monitoring provides an independent check on site-specific monitoring. 

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