Slide 1: APPLICATIONS OF
DIFFERENCE EQUATIONS IN BIOLOGY RECURRENCE RELATION : RECURRENCE RELATION "Difference equation" redirects here. It is not to be confused with differential equation.
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms.
The term difference equation sometimes refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation.
An example of a recurrence relation is the logistic map:
with a given constant r; given the initial term x0 each subsequent term is determined by this relation.
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n. HOW DO THEY APPLY IN BIOLOGY : HOW DO THEY APPLY IN BIOLOGY Some of the best-known difference equations have their origins in the attempt to model population dynamics. For example, the Fibonacci numbers were once used as a model for the growth of a rabbit population.
The logistic map is used either directly to model population growth, or as a starting point for more detailed models. In this context, coupled difference equations are often used to model the interaction of two or more populations. For example, the Nicholson-Bailey model for a host-parasite interaction is given by
with Nt representing the hosts, and Pt the parasites, at time t. THE BEE ANCESTRY CODEUsing FIBONACCI NUMBERS : THE BEE ANCESTRY CODEUsing FIBONACCI NUMBERS Fibonacci numbers also appear in the description of the reproduction of a population of idealized honeybees, according to the following rules:
1)If an egg is laid by an unmated female, it hatches a male or drone bee.
2)If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2. (This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.) LOGISTIC MAP : LOGISTIC MAP Slide 6: The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. Mathematically, the logistic map is written as where:
is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)
r is a positive number, and represents a combined rate for reproduction and starvation.
This nonlinear difference equation is intended to capture two effects.
reproduction where the population will increase at a rate proportional to the current population when the population size is small.
starvation (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population. LOGISTIC DIFFERENTIAL EQUATION : LOGISTIC DIFFERENTIAL EQUATION The logistic function is the solution of the simple first-order non-linear differential equation
where P is a variable with respect to time t and with boundary condition P(0) = 1/2. This equation is the continuous version of the logistic map.
The qualitative behavior is easily understood in terms of the phase line: the derivative is 0 at P = 0 or 1 and the derivative is positive for P between 0 and 1, and negative for P above 1 or less than 0. This yields an unstable equilibrium at 0, and a stable equilibrium at 1, and thus for any value of P greater than 0 and less than 1, P grows to 1.
The solution is given by: Modeling Population Growth : Modeling Population Growth A typical application of the logistic equation is a common model of population growth, originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth.
Letting P represent population size and t represent time, this model is formalized by the differential equation:
where the constant r defines the growth rate and K is the carrying capacity. Slide 9: In the equation, the early growth rate is modeled by the first term +rP. The value of the rate r represents the proportional increase of the population P in one unit of time. Later, as the population grows, the second term, which multiplied out is −rP2/K, becomes larger than the first as some members of the population P interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the bottleneck, and is modeled by the value of the parameter K. The competition diminishes the combined growth rate, until the value of P ceases to grow (this is called maturity of the population).
Dividing both sides of the equation by K gives Slide 10: Now setting gives the differential equation
For r=1 we have the particular case with which we started.
In ecology, species are sometimes referred to as r-strategist or K-strategist depending upon the selective processes that have shaped their life history strategies. The solution to the equation (with being the initial population) is
Which is to say that K is the limiting value of P: the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value P(0) > 0, also in case that P(0) > K Time-Varying Carrying Capacity : Time-Varying Carrying Capacity Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying: K(t) > 0, leading to the following mathematical model:
A particularly important case is that of carrying capacity that varies periodically with period T:
A typical value of T is one year: in such case K(t) reflects periodical variations of weather conditions.
Another interesting generalization is to consider that the carrying capacity K(t) is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation, which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth punctuated growth or alternation to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death. In Medicine: Modeling Of Growth Of Tumors : In Medicine: Modeling Of Growth Of Tumors Another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors Denoting with X(t) the size of the tumor at time t, its dynamics are governed by:
which is of the type:
where F(X) is the proliferation rate of the tumor.
If a chemotherapy is started with a log-kill effect, the equation may be revised to be
where c(t) is the therapy-induced death rate. i.e. if the average therapy-induced death rate is greater than the baseline proliferation rate then there is the eradication of the disease. LUDWIG VON BERTALANFFY : LUDWIG VON BERTALANFFY Karl Ludwig von Bertalanffy was an Austrian-born biologist known as one of the founders of general systems theory (GST). GST is an interdisciplinary practice that describes systems with interacting components, applicable to biology, cybernetics, and other fields. His mathematical model of an organism's growth over time, published in 1934, is still in use today.
Von Bertalanffy grew up in Austria and subsequently worked in Vienna, London, Canada and the USA.
Today, Bertalanffy is considered to be a founder and one of the principal authors of the interdisciplinary school of thought known as general systems theory. According to Weckowicz (1989), he "occupies an important position in the intellectual history of the twentieth century. His contributions went beyond biology, and extended into cybernetics, education,history, philosophy, psychiatry, psychology and sociology. Some of his admirers even believe that this theory will one day provide a conceptual framework for all these disciplines". Spending most of his life in semi-obscurity, Ludwig von Bertalanffy may well be the least known intellectual titan of the twentieth century. The Individual Growth Model : The Individual Growth Model The individual growth model published by von Bertalanffy in 1934 is widely used in biological models and exists in a number of permutations.
In its simplest version it is expressed as a differential equation of length (L) over time (t):
when is the von Bertalanffy growth rate and the ultimate length of the individual. The Dynamic Energy Budget theory provides a mechanistic explanation of this model in the case of isomorphs that experience a constant food availability. The inverse of the von Bertalanffy growth rate appears to depend linearly on the ultimate length, when different food levels are compared. The intercept relates to the maintenance costs, the slope to the rate at which reserve is mobilized for use by metabolism. The ultimate length equals the maximum length at high food availabilities. Slide 15: BERTALANFFY MODULE
To honor Bertalanffy, ecological systems engineer and scientist Howard T. Odum named the storage symbol of his General Systems Language as the Bertalanffy module
GENERAL SYSTEM THEORY (GST)
The biologist is widely recognized for his contributions to science as a systems theorist; specifically, for the development of a theory known as General System Theory (GST). The theory attempted to provide alternatives to conventional models of organization. LOTKA–VOLTERRA EQUATION : LOTKA–VOLTERRA EQUATION The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They evolve in time according to the pair of equations:
x is the number of its prey (for example, rabbits);
y is the number of some predator (for example, foxes);
and represent the growth of the two populations against time;
t represents the time; and
α, β, γ and δ are parameters representing the interaction of the two species. History of the equations: : History of the equations: The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in 1910. This was effectively the logistic equation, which was originally derived by Pierre François Verhulst. In 1920 Lotka extended, the model to "organic systems" using a plant species and a herbivorous animal species as an example and in 1925 he utilised the equations to analyse predator-prey interactions in his book on biomathematics arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic independently investigated the equations in 1926. Physical Meanings Of The Equations : Physical Meanings Of The Equations The Lotka-Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:
1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the prey populations.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species and the genetic adaptation is sufficiently slow.
As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping. Slide 19: PREY
When multiplied out, the prey equation becomes:
The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.
With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon. Slide 20: PREDATORS
The predator equation becomes:
In this equation, δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). γy represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.
Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death. Slide 21: Solutions to the equations
The equations have periodic solutions and do not have a simple expression in terms of the usual trigonometric functions. However, a linearization of the equations yields a solution similar to simple harmonic motion with the population of predators following that of prey by 90°. Slide 22: An example problem
Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40 cheetahs, one can plot the progression of the two species over time. The choice of time interval is arbitrary. Slide 23: DYNAMICS OF THE SYSTEM
In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low the prey population will increase again. These dynamics continue in a cycle of growth and decline.
Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0.
When solved for x and y the above system of equations yields
Hence, there are two equilibriums.
The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters, α, β, γ, and δ. Slide 24: Stability of the fixed points
The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives, while the other fixed point requires a slightly more sophisticated method.
The Jacobian matrix of the predator-prey model is REPLICATOR EQUATION : REPLICATOR EQUATION In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, in that it allows the fitness landscape to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. The replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies. Slide 26: EQUATIONAL FORMS
The most general continuous form is given by the differential equation
is the proportion of type i in the population, is the vector of the distributionof types in the population, is the fitness of type i(which is dependent on the population), and
is the average population fitness (given by the weighted average of the fitness of the types in the population). Since the elements of the population vector x sum to unity by definition, the equation is defined on the n-dimensional simplex.
The replicator equation assumes a uniform population distribution; that is, it does not incorporate population structure into the fitness. The fitness landscape does incorporate the population distribution of types, in contrast to other similar equations, such as the quasispecies equation. Slide 27: In application, populations are generally finite, making the discrete version more realistic. The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing. Note that the continuous form can be obtained from the discrete form by a limiting process.
To simplify analysis, fitness is often assumed to depend linearly upon the population distribution, which allows the replicator equation to be written in the form:
the payoff matrix A holds all the fitness information for the population: the expected payoff can be written as
and the mean fitness of the population as a whole can be written as Slide 28: Relationships To Other Equations
The continuous replicator equation on types is equivalent to the Generalized Lotka–Volterra equation in dimensions. The transformation is made by the change of variables
where is the Lotka–Volterra variable.