The aim of the paper is to give examples of apparent disappearance phenomenon which is typical for microbiological systems. It is well known that canards play an important role in the theory of mixed−mode oscillations consisting of large amplitude, and followed by small amplitude oscillations. In a sense we consider limiting cases of mixed−mode oscillations, when large amplitude oscillations are followed by vanishingly small amplitude oscillations.
The aim of the paper is to give illustrative examples of apparent disappearance phenomena which are typical for microbiological systems. Our interest in this topic was prompted by a recent trip to Israel made by the author several years ago. During the visit he was told about a group of Bedouins who, together with camels, were drowned in a flash flood in the desert. The following day he toured an old cemetery in Jerusalem that contained the remains of some English soldiers. The majority of the soldiers had died in a similar flash flooding event. In both cases, the victims had set up camp on a dry river bed. The river bed remained dry for long periods of time throughout the year. However, this normal state disappeared for a few hours when the catastrophic flooding occurred. Similar behaviour occurs in the illustrative model of a hydrodynamical system shown in the figure below, in which the lower reservoir is empty apart from a very short time interval where it is full.
The second reservoir also empties through a siphon. A small inflow causes the water level in the first reservoir to rise slowly until the U-turn in the siphon is reached. The water is then very quickly transferred to the lower reservoir due to the siphoning effect, filling it almost instantly. The second siphon then causes the lower reservoir to empty at an equally fast rate.
Strange as it may seen, the analogous situations appear in systems of diversified natures, such as in the case for ``predator-prey’’ systems in microbiology. In this paper we discuss some examples and mathematical models of the apparent disappearance phenomenon in population dynamics. in particular, the turning points, are isolated on S. Let, for simplicity, ( , 0) > 0, and ( , 0) < 0, for < ; ( , 0) > 0, for > .
Trajectories which at first pass along the attractive part of S and then continue for a while along the repulsive part of S are called canards or duck-trajectories. In the case that a turning point is unique, S is a canard and it is a longest canard.
Otherwise, this canard may be bounded by neighbouring turning points (or points at which ( , 0) = 0).
In many papers devoted to canards this term is associated with periodic trajectories
[
We will introduce some types of canards [
In the above cases properties of attraction or repulsion can be viewed as exponential attraction and exponential repulsion by analogy with exponential stability. Using an analogy with the asymptotic stability, we introduce the following notion.
Trajectories which are canards but in which attraction and repulsion are not exponential are called feeble canards.
If on one side of the trajectory it looks like a canard, but on the other side it looks like a false canard, we'll say that this trajectory is a two−faced canard.
It is easy to extend these concepts to the case of vector variables x and y. If, in
particular, x is a vector and y is a scalar, then we consider the subspace = 0 as a
slow manifold. If it contains repulsive and attractive parts, it is called a black swan
(attractive/repulsive invariant manifold) [
It should be noted that attractive/repulsive invariant manifolds can be considered for the case of small ε just as for the case ε = 1. The essence of the effect of apparent disappearance is as follows: variable y has vanishingly small amplitude for most of the time. This is due to the fact that trajectories of the system under consideration asymptotically approach an attractive part of the curve (surface) y=0 and escape from it repulsive part with a substantial time delay.
Consider the system = ( , ), ε = ( , ); (2) where ≥ 1, ( , ) is positive for < 1 and ( , ) < 0 for > 1, and moreover, that f increases to infinity when → ∞. Let also ( , 0) be negative for < 0 and positive for > 0. Then = 0 is a canard; this canard is feeble if > 1, and it is a two−faced canard if = 2,4, … . Suppose that this system has periodic solutions for any small ε > 0, and that the graph of this solution crosses the interval = {( , 1): − δ < < 0}, where δ is a positive constant. Then for small ε the qualitative features of this periodic orbit are predetermined, and may be divided in the six stages. S1 The trajectory falls down almost vertically, until it will be of order ε ln ε. S2 It will continue approach zero moving simultaneously to the right until ( ) becomes equal to 0. The rate of approaching zero by ( ) is exponential, with an exponent ~ -1/ε, if = 1 and convergence is polynomial if > 1. S3 The trajectory will still follow very closely the x-axis, but ( ) will increase until it becomes of order ε ln ε; this stage is essentially symmetric with S2. S4 The trajectory will go up almost vertically, until ( ) = 1 (“symmetrically” to stage S1). S5 The solution will go up (rapidly), simultaneously drifting faster and faster to the left until ( ) is equal to 0. S6 At the final stage the trajectory goes down and to the left until it cross again the interval. This stage is essentially symmetric with Stage S5.
The dynamics of ( ) exhibits the essential features of theapparent disappearance phenomenon, because all stages, except for S2 and S3, are fast. It is easy to give examples of equations (2) where this scenario happens, that is where the above mentioned periodic trajectories exist. For instance, if is even in : ( , ) = (− , ), and is odd: ( , ) = − (− , ), then any solution is periodic. Figure 2(b), right graphs sample trajectories of an illustrative system = 1 − , ε = ; (3) Here the trajectories can be written explicitly as 2 + 2ε( − ln ) = , where > 0 is a parameter.
The crucial question is the following: are there many real world phenomena which are consistent with this scheme, or with its suitable amendments?
The purpose of the paper is to demonstrate that the answer is in the affirmative.
The classic Lotka-Volterra Equations were suggested to model population
dynamics of a predator--prey system. They are [
Hoffmann [
The modified Lotka-Volterra equations are therefore: = (α − βΦ ), Φ = Φ (−γ + δ ); we will assume the inequality > 1.
Loosely speaking the equations (6) mean that the “effective size” of the phage population differs from its “physical size”, and this effective size is proportional to the power of the physical size. A naive explanation may be as follows: in the traditional Lotka-Volterra equations (5) and (6), it is assumed that the important meetings are when one predator meets one prey. With the power of two, which was among the best fits in the modified equations, we can hypothesize that unlike the traditional system, it is the meeting of two phages and one bacteria that are important. Correspondingly, the power Φ in (6) indicated that that are ``hunting−teams'' of phages which are effective. This sounds strange for a fractional Φ, but such ideas are not unusual in mathematics.
This system also has a unique positive equilibrium:
1 Φ∗ = (α/β) . (7) ∗ = γδ, Returning to equations (6), we can look for a constant of motion for the system. We do this by dividing the first equation (6) by the second to obtain (− log( ( )+ δ ( ))+ (αΦ( )1− )/( − 1)+ β Φ( )) = . for any positive solution ( ( ), Φ( )) of (5). (4) (6)
Thus, ( , Φ) = (−γ log( ( ) + δ ( )) + (αΦ( )1− ) −1 + β Φ( )) is a conserved quantity, in other
words, the first integral, for the motion of trajectories in this system. The value of the constant is determined by the initial conditions. Rewriting system (6) as = (αΦ∗ − Φ ), we can see the direction vector field in the strictly positive quadrant. as → 0, or Φ → 0, or → ∞, or Φ → ∞, the trajectories system could not expand (8) (9)
Since to infinity. > 1,
lim ( , ) = ∞
Finally, since the restriction ( ) = ( , Φ∗) is strictly monotone for 0 <
< ∗ , around the equilibrium ∗ = γ δ ,
1 Φ∗ = (α/β) . with > 0.
< we conclude that all trajectories of (6) are periodic, and they orbit anti−clockwise
Let us consider the modified Lotka-Volterra equations (6), when one of the variables (
or Φ) is significantly faster than the other. We can use singularly perturbed differential systems for modelling such phenomena. The biologically relevant case is “Fast Phages - Slow Bacteria”, and the corresponding system is = (α − βΦ +1), ε
= Φ +1(−γ + δ ), Φ Proposition 1. The trajectory Φ = 0
is a feeble canard with the attractive part = ∗ and the repulsive part = ∗ for positive and .
We can observe here the effect of apparent disappearance of phages: they almost disappear for most of the time; however the phages population “explodes” for some short time intervals, see Fig. 3. A curious point is that this feeble canard is two-faced if = 2, 4, 6, . .. , but that is irrelevant because we are interested in the behavior of our system for positive values of Φ, only.
We consider also the “Fast Bacteria - Slow Phages” case with the corresponding equations ε = (α − βΦ +1), Φ = Φ +1(−γ + δ ).
Note that this case is not interesting from the biological point of view. Proposition 2. The trajectory = 0 is a canard with the attractive part > 1 1 ∗ = ( / ) and the repulsive part < ∗ = ( / ) .
In this case we observe the effect of the apparent disappearance of bacteria, see Fig. 4.
The modification of the famous Lotka-Volterra equations is considered in the paper. This modification has been recently suggested to model the structure of marine phage populations, who are the most abundant biological entities in the biosphere. The paper attempts to realize the following intentions: to make some methodical remarks concerning this modification; to discuss new types of canards which arise naturally in this context; to present results of some numerical experiments.
This work is supported in part by the Russian Foundation for Basic Research (grants 13-01-97002-p) and the Ministry of education and science of the Russian Federation in the framework of the implementation of Program of increasing the competitiveness of SSAU for 2013−2020 years.