Qualitative modelling in ecology

Examples of Applied Qualitative Techniques. Chapter 6 Qualitative modelling in ecology Miguel Toro 1 Introduction In this example a non-linear system is examined. It can be used as a model of the growth of two species. The model tries to capture, in the simplest way, dependencies of growth rate on food supply and negative effects of over crowding. In the same way, dependencies of the growth rate of predators on the quantity of preys are captured. Instead of explicit formulas for the equations, certain qualitative assumptions are made regar�ing the form of the equations. These assumptions show the qualitative k.nowledge about the system that should be taken. Qualitative analysis and bifurcation theory can be applied to obtain the long-run qualitative behaviour of the trajectories of the system. 2 Specifying a model The system to model, is composed by preys and predators. The preys depend on the environment and predators depend on the preys. Every population has a birth rate and a death rate. The growth rate is defined as the birth rate minus the death rate. In the following x stands for l?rey and y stands for predators. The k.nowledge about the variables in the model and relations between them are given by the following assumptions: The growth rate of preys depends only on the population of preys. Let fl(x) be the growth rate of the prey. It is plausible to assume that there is a limiting population x 1 such that fl(xl)=O, fl(x)<O, x>xl. The growth rate is assumed to be positive for small populations fl(O)=a>O. Take into account positive social phenomena, for example, a medium size population may be better organised to obtain food than a small one. This implies that there is a value xO of the population where fl has a local maximum. So see that fl(xO)=b, O<xO<xl, b>a. Let us study the consequences in the behaviour of the aspect of fl. The properties of fl are: { O < xO < x l,O < a < b fl = fl(O) = a,fl(xl) = O fl(xO) = b,fl'(xO) = O The prey population is the total food supply for the predator. The growth rate of the predators depends on the prey populations. Let f2(x) be the growth rate of predators. Let us assume that there is a value x2 of x such as f2(x2) =O and that f2 is a monotone increasing function. It is known that for large sizes of prey population, the growth rate has a value b. So f2( oo )=b. The properties of f2 are: Examples of App!ied QualitativeTechniques Chapter 6 {x2 > O,b > O /2 = f2(x2) = O /2(00) = b Let us assume that the number of prey killed by each predator is a monotone increasing function of prey number. Let D(x) be this number. Then D(O)=O, D(oo)=d and D(x) = e x for small and medium values of x. { d > 0,e > O f3(x) = /3(0) = 0,/3(00) = d f3(x)-= ex far small x Other details regarding ecological model can be found in [ART93]. 3 Qualitative model Ideas from [KUI86] are here used to represent the qualitative k.nowledge of the systems. This k.nowledge is represented by a set of variables, a set of important values of these variables and set of functional constraints between them. Monotone and nonmonotone functional constraints are .used, and the derivative operator and arithmetic operators do. A list of couples of points to specify the properties of functional constraints is here g1ven. The qualitative model of the system is a>O b>a,c>O,d>O O< xO,xO < xl f 1: (-00,-00 ), (0,a ), (xO, b ), ( xl,O), ( +00,-00) f 2: (-00,-00 ), ( x2,0), ( +oo, e) f 3: (-00,-00 ), (0,0), ( +oo,d) dx -=x fl(x)-f3(x) y dt dy -=f2(x)y dx 109 s • 5 ' s 8 I í t o ' s ^ . t o a I 8 B 2 S 3 O ^ 3 e n o ^ ^ o í o p ^ P B a - a B o n > 3 . c u o > i 5 2 g ^ B 2 ^ p ^ o e n P P a P l ^ u l i I i 3 ^ 2 - a ^ ^ < ^ g l ^^ c ^ ^ C / l ^ . ( T i • / * ^ B ^ 2 . í ? p . 5 e n P B e n o w o j A 3 e n O e n PP e n P e n 3 O B - I - + ; P B e n b ^ 2 O O 3 g : p b. 3 e n ' - O o u ^ ^ a 3 . w a r r - c c r . e n 2 - B ^ p ^ . 2 a . B 3 - a ; p a 3 c r a 3 3 o o O O 3 e n a O 3 B e n 3 c e e n 0 3 e n e n c ^ • o í e n i " 03 O e n e n p P O 1 - 1 e n o B e n e n e n X 3 X e n 03 O I p e n P o b ^ U - 3 S a ¡ I ^ p ^ 3 * e n B 3 g O > - t " < o ^ o e n 3 - & g E S . o p o ^ o s ^ 3 2 ^ 2 . e n o í 2 . 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