Chapter 1
Qualitative Methods for Studying Dynamic Models of Biological Processes

The functioning of the integrated biological system is a result of interactions of its components in time and space. Elucidation of the principles of regulation of such a system is a problem that can be solved only with the use of correctly chosen mathematical methods.

The kinetics of biological processes includes the time-dependent behavior of various processes proceeding at different levels of life organization: biochemical conversions, generation of electric potentials on biological membranes, cell cycles, accumulation of biomass or species reproduction, interactions of living populations in biocommunities.

1.1 General Principles of Description of Kinetic Behavior of Biological Systems

The kinetics of a system is characterized by a totality of variables and parameters expressed via measurable quantities, which at each instant of time have definite numerical values.

In different biological systems, different measurable values can play the role of variables: those are concentrations of intermediate substances in biochemistry, the number of microorganisms or their overall biomass in microbiology, the species population number in ecology, membrane potentials in biophysics of membrane processes, etc. Parameters may be temperature, humidity, pH, electric conductance of membranes, etc.

This is sufficient to construct a general mathematical model representing a system of n differential equations:

where c1(t), …, cn(t) are unknown functions of time describing the system variables (for example, substance concentrations); dci/dt are rates of changes of these variables; fi are functions dependent on external and internal parameters of the system. A comprehensive model of type (1.1) may contain a large number of equations, including nonlinear ones.

Many essential questions concerning the qualitative character of the system behavior, in particular, stability of stationary states and transition between them, oscillation modes and others, can be solved using methods of the qualitative theory of differential equations. These methods permit revealing important general properties of the model without determining explicitly the unknown functions c1(t), …, cn(t). Such an approach gives good results when analyzing the models that consist of a small number of equations and reflect the most important dynamic features of the system.

The key approach in the qualitative theory of differential equations is to characterize the state of the system as a whole by variables c1, c2, …, cn, which they aquire at each instant of time upon changing in accord with (1.1). If the values of variables c1, c2, …, cn are put on rectangular coordinate axes in the n-dimensional space, the system state will be described by some point M in this space with coordinates M(c1, c2, …, cn). The point M is called a representation point.

The change in the system state is comparable to the displacement of the point M in the n-dimensional space. The space with coordinates c1, c2, …, cn is a phase state; the curve, described in it by the point M, is a phase trajectory.

1.2 Qualitative Analysis of Elementary Models of Biological Processes

Let us consider qualitative methods of studying such systems represented as a system of two independent differential equations (the right-hand parts do not depend explicitly on time), that can be written as:

Here P(x, y) and Q(x, y) are continuous functions, determined in some range G of the Euclidean plane (x and y are Cartesian coordinates) and having continuous derivatives not lower than the first order.

The range may be both unlimited and limited. When variables x and y have a certain biological meaning (substance concentrations, species population number), some restrictions are usually superimposed on them. First of all, biological variables cannot be negative.

Accept the coordinates of the representation point M0 to be (x0, y0) at t = t0.

At every next instant of time t, the representation point will move in compliance with the system of equations (1.2) and have the position M(x, y), corresponding to x(t), y(t). The set of points on the phase plane x, y is a phase trajectory.

The character of phase trajectories reflects general qualitative features of the system behavior in time. The phase plane, divided in trajectories, represents an easily visible “portrait” of the system. It allows grasping at once the whole set of possible motions (changes in variables x, y) corresponding to the initial conditions. The phase trajectory has tangents, the slopes of which in every point M(x, y) equals the derivative value in this point dy/dx. Accordingly, to trace a phase trajectory through point M1(x1, y1) of the phase plane, it is enough to know the direction of the tangent in this point of the plane or the value of the derivative

To this end, it is required to have an equation with variables x, y and without time t in an explicit form. For that, let us divide the second equation in system (1.2) by the first one. The following differential equation is obtained

which is frequently much more simple than the initial system (1.2). Solution of equation (1.3) y = y(x, c) or in an explicit form F(x, y) = C, where C is the constant of integration, yields a family of integral curves — phase trajectories of system (1.2) on the plane x, y.

But generally, equation (1.3) may have no analytical solution, and then integral plotting should be done using qualitative methods.

Method of Isoclinic Lines. The method of isoclinic lines is typically used for qualitative plotting of a phase portrait of a system. In this case, lines, which intersect the integral lines at a certain angle, are plotted on the phase plane. The analysis of a number of isoclinic lines can show the probable course of the integral lines.

The equation of isoclinic lines can be obtained from equation (1.3). Suppose dy/dx = A, where A is a definite constant value. The value of A is a slope of the tangent to the phase trajectory and, consequently, can have values from –∞ to +∞. Substituting the A value instead of dy/dx in (1.3), we get the equation of isoclinic lines:

By giving different definite numeric values to A, we obtain a family of curves. In any point of each of these curves, the tangent slope to the phase trajectory, passing through this point, is the same value, namely the value of A, which characterizes the given isoclinic line.

Note that in the case of linear systems, i.e. systems of the type

isoclinic lines represent a bundle of straight lines, passing through the origin of coordinates:

Singular Points. Equation (1.3) determines directly the singular tangent to the corresponding integral curve in each point of the plane. Exclusion is the point of intersection of all isoclinic lines , at which the tangent direction is indefinite, because in this case the value of the derivative is ambiguous:

The points, in which time derivatives of variables x and y turn concurrently to zero

and in which the direction of tangents to integral curves is indefinite, are singular points. The singular point in the equation of phase trajectories (1.3) complies with the stationary state of system (1.2), because the rates of changes of variables in this point are equal to zero, and its coordinates are stationary values of variables .

For a qualitative study of a system, it is often possible not to go beyond plotting only some isoclinic lines on the phase plane. Of special interest are the so-called basic isoclinic lines: dy/dx = 0 is the isoclinic line of horizontal tangents to phase trajectories, the equation of which is Q(x, y) = 0, and the isoclinic of vertical tangents dy/dx = ∞, which is in line with equation P(x, y) = 0.

The plotting of the basic isoclinic lines and the determination of their intersection point, the coordinates of which satisfy the following conditions

gives the intersection point of all isoclinic lines on the phase plane. As mentioned above, this point is a singular point and corresponds to the stationary state of the system (Fig. 1.1).

Figure 1.1. The stationary state is determined by the point of intersection of the basic isoclinic lines.

Figure 1.1 demonstrates the case of one stationary point of intersection of basic isoclinic lines of the system. The figure shows directions of the tangents...