1. Introduction
It is a fact that species coexistence in nature is closer to being the rule rather than the exception. This empirical observation is at odds with the classical ecological theory established in the Competitive Exclusion Principle: two species competing for the same limited resource cannot coexist at constant population values. However, ref. [
1] posed the so-called Paradox of the Plankton questioning how so many species of marine plankton coexist in a rather homogeneous medium with limited food, seemingly contradicting Gause’s rule of competitive exclusion. This is just one of many examples illustrating how puzzling coexistence is [
2]. Since then, much effort has been invested in conciliating observational and theoretical knowledge, describing different frameworks and mechanisms that explain such a contradiction; see [
3] or [
4] for recent reviews.
Competition can be modeled in different ways [
5]. In exploitative (scramble) competition, individuals can take a portion of the resource, which appears explicitly in the model. In contrast, in interference (contest) competition, one, and only one, of the contenders will keep the whole resource, which is not explicit in the equations. The prototype models of interference competition are [
6] (the competition counterpart of the Lotka–Volterra predator–prey model) and [
7], the first formulated with ordinary differential equations and the latter with difference equations. Both of them were based on laboratory experiments (two species of fungi,
Saccharomyces cerevisiae and
Schizosaccharomyces kefir, and two species of flour beetle,
Tribolium castaneum and
Triblium confusum, respectively) and are considered to support the Competitive Exclusion Principle.
The above-mentioned models are dynamic equivalents and exhibit four competition outcomes (apart from the extinction of both species): either species 1 or species 2 wins regardless of the initial number of individuals; species coexistence occurs no matter the initial number of individuals; or the so-called priority effects occur (one species will go extinct, depending on the initial number of individuals). The models share a key feature: their nullclines (the zero-growth curves) are straight lines, which implies the existence of at most one coexistence equilibrium point in the interior of the positive cone. Observational experiments [
8,
9] show that nullclines can be different from straight lines. In [
10], the authors underline that Lotka–Volterra models exhibit per-capita growth rates that are linear functions of density and may bias their predictions (see references therein). Despite this empirical evidence and these theoretical reasons, interference competition models have remained the same for nearly 80 years, and effort has been put into exploitative competition [
4]. Recently, several authors have resumed research on interference competition by incorporating into the classical Gause model different features, such as herd behavior [
11,
12,
13], inter-species interference [
3], and group defense [
14] behavior.
Interestingly, none of the above works (with the exception of [
3]) accounted for
the way interference takes place, despite being expansions of the classical
interference competition model. In [
2], the authors discussed the matter of (co)existence by stating the following:
“To fully appreciate how such large numbers of species can coexist, we need to have a thorough understanding of the dynamics of their populations and, in particular, of how individuals within a species’ population interact with each other, and with populations of other species. Indeed, the coexistence of individuals within a population is intriguing, given that interactions within species can in fact be stronger than interactions among the populations of different species.”
Also, Ref. [
2] pointed out that clustering is a coexistence mechanism: the strongest competitor faces mainly conspecifics, and the same happens to the weaker competitor.
The above ideas strongly suggest that it is important to account for the balance between intra-species interference and inter-species interference in order to better understand species coexistence. We follow the works [
15,
16] on predator–prey models to fully incorporate the effect of individual interference on species competition into the classical Gause competition model. When individuals of different species want the same resource, it may take some time to “decide” which of them will obtain the resource, as explained in [
3]. Also, when a few individuals of both species want the same resource, individuals interfere with not only heterospecifics but also conspecifics. We incorporate the latter feature into the model analyzed in [
3].
The new model that we present here displays a different feature from those observed in previous models: the global species coexistence region is larger than in all the preceding models, which reveals that interference with homospecifics is key to coexistence. We found the dynamic scenarios described in [
3] (those allowed by the classical model and conditional coexistence in favor of one or the other species and conditional coexistence), along with bi-coexistence in [
14], although the biological reasons leading to each dynamic scenario are different. We derived a closed formula describing the conditions for the above-mentioned features in particular but meaningful settings.
The manuscript is organized as follows. In
Section 2, we derive in detail the above-mentioned model. In
Section 3, we assume that intra- and inter-species interference only affects one of the competing species. In
Section 4.1, we consider just intra-species interference in both species. In
Section 4.2, the full model is analyzed under symmetric competition assumptions. Note that the analytical description of the full model without further restrictions is algebraically feasible but biologically uninterpretable. Finally, we discuss the mathematical results from an ecological viewpoint in
Section 5. The mathematical technicalities are gathered in the
Appendix A.
2. The Beddington–DeAngelis Competition Model and Preliminary Results
The base model is nothing but the classical Gause competition model [
6]:
where
stands for the number of individuals of species
,
is the intrinsic growth rate of species
, and
the coefficient accounting for intra-species (
) and inter-species (
) competition for
. We considered the emergent carrying capacities formulation as in [
17,
18] rather than the usual carrying capacities [
19].
The main feature of the classical model (
1) is that the per capita growth rate decreases linearly with
and
(
), i.e.,
As we pointed out in [
3,
14], the hypothesis that the competitive pressure exerted by a fixed number of individuals of species
j on species
i is the same
regardless of the number of individuals of species
i does not make sense in reality. For instance, 100 individuals of species
j would not affect in the same way, say, 20 or 2000 individuals of species
i. At the same time, in
Section 1, we justified the interest in disentangling intra-species from inter-species interference when competing.
This task was addressed by Beddington [
15] and DeAngelis [
16] for predator–prey models. We followed their works to build a competition model that separates the contribution of inter-species interference as well as mutual interference when competing.
Let us consider species
i. In the classical model, species interactions do not take time, so that individuals are active for the whole time interval
T considered, i.e.,
. In [
3], we assumed that inter-species competition takes time, so that
plus the time spent facing species
j. We further assume the intra-species interference of individuals of species
i when competing with species
j. That is to say, a few individuals of each species seek the same resource, and only one individual can obtain it. Then, there is also intra-species interference when disputing a given resource with heterospecifics. Specifically, we partition
T as follows:
where
is the time for which each individual of species
is active;
and
stand for the average time taken by an interaction between one individual of each or the same competing species, respectively; and
and
are the numbers of competitors from species
and
i, respectively, that become extinct due to the interference of
a single individual of species
. Indeed,
can be expressed as follows:
where
is the total number of individuals of species
j, and
is the product of the resource finding rate of species
j times the probability of meeting a competitor of species
. Also, the number of encounters between individuals of species
is given by
where
is the rate of encounters between equals within species
and is related to both their speed of movement and the range at which they sense each other. Note that
is used instead of
since each individual cannot interfere with itself.
Substituting (
4) and (
5) into (
3) and solving the corresponding equation in
yields
Recall that
is the number of individuals of
j ruled out by a single individual of species
i that is given by (
4). Then, the total number of individuals of
j ruled out by species
i is given by
Plugging (
7) into System (
1) and relabeling the coefficients as
,
, and
yields
Therefore, and account for the effect of interference between heterospecifics and homospecifics on species i when both competing species interact.
Remark 1. Competition is related to but different from interference. In intra-species competition (), individuals of the same species interact to obtain a resource in the absence of the other species. In inter-species competition (), individuals of different species interact to obtain a resource, and its effect on the other species is density-independent.
Interference models the density-dependent effects of inter-species competition. This takes place, for instance, when few individuals of each species compete for the same resource and face, annoy, disturb, etc., individuals of the other species () or the same species ().
Interference is also different from delayed effects (delay ODEs), as in the latter the effect of competition is felt with a certain delay, but the inter-species competition coefficient can remain density-independent.
We will refer to the term on the right-hand side in Equation (
8) as the
Beddington–DeAngelis competitive response. Let us rewrite System (
8) in a suitable way by setting
,
, and
. This yields
Note that , as it is the carrying capacity of species i in the absence of species . Thus, stands for the ratio to its carrying capacity. Furthermore, is the ratio of the carrying capacity of species j over .
System (
1) is a particular case of (
9) when
. Note that requiring
is mandatory to prevent the denominator of Equation (
8) from being equal to or less than zero at low population densities. We also assume
. Heuristically, on the one hand, the effect of inter-species interference is
, and it should not be larger than the whole carrying capacity. On the other hand,
should be comparable to
. Note that considering the above conditions for the parameter values, the right-hand side of System (
9) is smooth, ensuring the existence and uniqueness of solutions for any initial values in the non-negative cone
.
2.1. First Properties of System (9)
System (
9) is well behaved.
Theorem 1. Consider System (9). Then, the following are true: - 1.
The axes are forward-invariant.
- 2.
The solutions are bounded from above.
- 3.
The positive cone is forward-invariant.
Proof. We just provide an outline of the proof. For statement 1, setting
reduces System (
9) to the logistic equation. Statement 2 follows from the fact that the solutions of System (
9) are bounded from above by the solutions of a logistic equation (let
,
). Finally, statement 3 follows from 1 and 2. □
Theorem 2. Any solution of System (9) eventually converges monotonically to an equilibrium point. Proof. Direct calculations allow us to apply the results in [
20]. □
As a consequence, the flow of System (
9) strictly decreases outside the rectangle
on the positive cone. Thus, there is no equilibrium point for System (
9) in
.
2.2. Remarks on the Classical Competition Model
System (
1) is the scaffolding for Systems (
8) and (
9). Thus, let us present some basic known properties of the classical competition model (
1) [
19]. We consider System (
1) after the change of variables leading to System (
9). This is the same as the setting
,
in (
9).
The acronyms GAS and LocAS refer to the global and local asymptotic stability of a fixed point, respectively. Let
and
be the so-called semi-trivial equilibrium points that are fixed points of both Systems (
1) and (
9). It is known that
is GAS if
and
. Also,
is GAS if
and
. Moreover,
is an equilibrium point that is in the positive cone if either
or
. Indeed,
is GAS if
. When
becomes unstable. In this case,
and
are LocAS, and the unstable manifold of
is the separatrix of the corresponding basins of attraction.
Figure 1 displays the possible phase portrait configurations leading to each of the competition outcomes.
Figure 2 shows the corresponding bifurcation diagram with
and
as bifurcation parameters.
The above results point to 1 as a threshold value for comparison with the competitive strength
. Let us recall that the
competitive strength
relates the intra-species and inter-species competition coefficient and species intrinsic growth rates (growth rates must be taken into account to avoid meaningless results [
21]). From a different viewpoint,
compares the ratio of the competition effect of species
j on species
i over the intrinsic growth rate of species
i to the carrying capacity of species
j. Species
j cannot drive species
i to extinction if, and only if, the competitive strength
is less than 1.
We will refer throughout the manuscript to global coexistence when there is a single GAS interior equilibrium point, to coexistence when there is a LocAS interior equilibrium point, and to priority effects when , are LocAS and there exists an interior unstable equilibrium point with a separatrix passing through it that defines their basins of attraction.
3. The Beddington–DeAngelis Competitive Response for Just One Species
We analyze in this section the case where only species 2 suffers the effect of interference when competing. We then assume that species 1 does not interfere when competing with species 2,
, and when encountering individuals of species 1,
. Thus, the effect of the competition of species 2 on species 1 becomes density-dependent. That is to say, we analyze the system
System (
12) is a particular case of System (
9), so it is well behaved (see Theorem 1). We first draw conditions for the stability of the semi-trivial equilibrium points.
Theorem 3. Consider System (12) with and (recall that ). Then, the following are true: - 1.
The trivial equilibrium point is unstable.
- 2.
The semi-trivial equilibrium point is LocAS if .
- 3.
The semi-trivial equilibrium point is LocAS if
Proof. This follows by direct calculations and analyzing the sign of the eigenvalues of the Jacobian matrix calculated at and , . □
We next focus on the existence and stability of coexistence (positive) equilibrium points. These points are found from the nullclines of System (
12), that are given, respectively, by
Note that
is a straight line, as in the classical model (
1), and
is the rational function defined in (
14).
Equating
yields a second-degree equation:
where
It is a trivial task to solve Equation (
15) and present conditions for the existence of no, one, or two solutions. However, it is not straightforward to decide whether the solutions belong to the positive cone or not. We perform such an analysis in
Appendix A by combining Descartes’s rule of signs with analyzing the discriminant of the solution of Equation (
15), and Theorems 4 and 5 fully describe the possible outcomes of System (
12).
Figure 3 displays the configurations of the nullclines and the phase planes of System (
12) that are not allowed by the classical model.
Figure 4 shows the possible outcomes of System (
12) on the
plane.
Theorem 4. Consider System (12) with , , and . Then, the following are true: - 1.
Global coexistence. There exists a unique coexistence equilibrium point that is globally asymptotically stable if .
- 2.
Priority effects. Condition entails the existence of a unique coexistence equilibrium point that is unstable, while and are LocAS. Furthermore, is a saddle, and its stable manifold is a separatrix curve that divides the positive cone into two open regions, and , such that is on its boundary. Any solution with initial values in converges to , while any solution with initial values in converges to .
In what follows, two curves are key to complete the description of the outcomes of System (
12). These are the curves
, where
These curves arise from equating to zero the discriminant of the solution of Equation (
15) and solving for
in the resulting equation. Given fixed values of
,
, and
, these curves separate the regions of the
plane containing the real and the complex roots of Equation (
15).
Lemma 1. Consider (17). Then, the following are true: - 1.
at .
- 2.
.
- 3.
at .
- 4.
if, and only if, .
Proof. This follows from direct calculations. □
The following result describes the outcomes of System (
12) in terms of the relative size of
and
. Refer to
Section 4.1 for a biological interpretation.
Theorem 5. Consider System (12) with , , and . Then, the following are true: - 1.
Condition (see the bottom-left central panel in Figure 4) entails the following: - (a)
Species 2 wins. The equilibrium point is unstable, and is GAS to the positive cone for any - (b)
Species 1 wins. The equilibrium point is unstable, and is GAS to the positive cone for any
- 2.
Consider (see the bottom-right panel in Figure 4). This entails the following: - (a)
Species 2 wins. The equilibrium point is unstable, and is GAS to the positive cone for any - (b)
Species 1 wins. The equilibrium point is unstable, and is GAS to the positive cone for any - (c)
Coexistence or species 1 wins. Assume that are such that Then, is unstable, while is LocAS. In addition, two coexistence equilibrium points appear in the positive cone. That closer to is LocAS. The other is a saddle whose stable manifold is a separatrix defining the basins of attraction of the coexistence equilibrium and .
- 3.
Assume now that (see the top row of Figure 4). This entails the following: - (a)
Species 1 wins. The equilibrium point is unstable, and is GAS to the positive cone for any - (b)
Species 2 wins. The equilibrium point is unstable, and is GAS to the positive cone for any where is defined in Lemma 1.
- (c)
Coexistence or species 2 wins. Assume that are such that Then, is unstable, while is LocAS. In addition, two coexistence equilibrium points appear in the positive cone. That closer to is LocAS. The other is a saddle whose stable manifold is a separatrix defining the basins of attraction of the coexistence equilibrium and .
We leave the biological interpretation and discussion of the above results for
Section 5.3.1.
4. The Beddington–DeAngelis Competitive Response for Both Species
In this section, we focus on the complete model (
9). We already know that System (
9) is well behaved (see Theorem 1), and we seek the existence and stability of the positive equilibrium points, which are closely related to the stability of the semi-trivial equilibrium points.
Theorem 6. Consider System (9) with , . This entails the following: - 1.
The trivial equilibrium point is unstable.
- 2.
Consider the semi-trivial equilibrium points , . Then, the following are true:
- (a)
- (b)
Proof. This follows by accounting for the sign of the eigenvalues of the Jacobian matrix calculated at and , . □
As for the non-trivial equilibrium points of system (
9), one needs to find the intersection points of the nullclines, given by
in (
14) and
Thus, the positive equilibrium points are given by the positive roots of
where the explicit expression of the coefficients can be found in
Appendix B. The study of (
29) is feasible, but the conditions enabling one outcome or another are extremely numerous and not meaningful at all.
Thus, we deal with two particular but interesting cases. The study will be completed with numerical simulations.
4.1. Considering Only Intra-Species Interference
We now let
and
in System (
9), so that we neglect the effect of inter-species interference when competing and focus on the role of intra-species interference. Some of the results in
Section 4.1 were published in [
22]. We include herein the full version for the sake of completeness.
Theorem 6 deals with the semi-trivial equilibrium points. We next focus on the non-trivial equilibrium points.
Theorem 7. Consider System (9) with . Then, the following are true: - 1.
Global coexistence. There exists an equilibrium point in the non-negative cone that is GAS for any - 2.
Priority effects. There exists a saddle equilibrium point in the non-negative cone that is unstable for any Indeed, and are locally asymptotically stable, and the stable manifold of the positive (component-wise) equilibrium defines the basins of attraction of each semi-trivial equilibrium point.
Proof. The non-trivial equilibrium points are the solutions to the equation resulting from equating the nullclines of System (
9). Locating the equilibrium points in the positive cone follows the reasoning in
Appendix A. The stability of the non-trivial equilibrium points follows from Theorem 2 and the stability conditions of the semi-trivial equilibrium points proved in Theorem 6. □
The feasible non-trivial equilibrium points are the positive roots of the polynomial
where
Solving the discriminant of Equation (
32) for
yields two curves,
, that separate the regions in the
-
plane leading to no, one, or two equilibrium points.
Lemma 2. Consider the curves defined above. This entails the following:
- 1.
.
- 2.
.
- 3.
- 4.
- 5.
Proof. This follows from direct calculations. □
Theorem 8. Consider System (9) with . Then, consider the following: - 1.
Assume now that condition (36) holds. This entails the following (see the bottom-left panel in Figure 5): - (a)
Species 1 wins. is GAS whenever - (b)
Species 2 wins. is GAS whenever
- 2.
Instead, if condition (37) holds, this entails the following (see the panels in the first row in Figure 5): - (a)
Species 1 wins. is GAS if , as defined in (39). - (b)
Now, two different outcomes may occur in :
- i.
Species 2 wins. is GAS whenever , where - ii.
Conditional coexistence or species 2 wins. On the contrary, if , where then is locally asymptotically stable and is unstable. In addition, there exist two equilibrium points in the positive cone, one locally asymptotically stable and one unstable. The latter is a saddle equilibrium point whose stable manifold separates the basins of attraction of and the positive (coexistence) equilibrium point.
- 3.
Finally, if condition (38) holds, this entails the following (see the bottom-right panel in Figure 5): - (a)
Species 2 wins. is GAS if , as defined in (40). - (b)
Two different outcomes may occur in :
- i.
Species 1 wins. is GAS whenever , where - ii.
Coexistence or species 1 wins. On the contrary, if then is locally asymptotically stable and unstable. In addition, there exist two equilibrium points in the positive cone, one locally asymptotically stable and one unstable. The latter is a saddle equilibrium point whose stable manifold separates the basins of attraction of and the positive (coexistence) equilibrium point.
Proof. When equating the nullclines of System (
9) with
for
, we obtain a second-degree equation for
. The solution of such an equation is the
component of the equilibrium points of System (
9). Setting the discriminant of the solution of that equation equal to zero, the curves
and
are obtained. These curves bound the regions on the
plane where there are two, one, or no equilibrium points (that is, the algebraic equation has either real or complex solutions).
The signs of the coordinates of the equilibrium points are determined by using Descartes’ rule of signs. The number of equilibrium points inside the non-negative cone, in addition to the stability of the semi-trivial equilibrium points (Theorem 2), yields the stability of the non-trivial equilibrium points. □
4.2. Symmetric Competition
Symmetric competition may take place, for instance, when individuals of different species display similar phenotypic traits [
23]. We introduce this feature in System (
9) by setting
Assumption (
45) is a strong one. We are interested in the qualitative outcomes of the model. Note that the stability of hyperbolic equilibrium points is robust under small (enough) perturbations in the parameters of the system. Thus, the results achieved also hold for coefficients close to the perfect symmetry settings stated above.
A first consequence of the symmetry assumptions (
45) is that there exists either one equilibrium or three equilibrium points in the positive cone. In the former case, there is either coexistence or priority effects. In the latter case, the outcomes are either global coexistence through two positive stable equilibrium points or tri-stability (coexistence or priority effects).
The symmetry conditions (
45) simplify the fixed-point Equation (
29) so that a computer algebra system (we used the commercial software Mathematica [
24]) can find the explicit expression of the solutions:
where
and
Note that
implies
so that
and
always exist. Indeed, both lie on the line
:
on the positive semi-line, and
on the negative side.
and
can be either complex or real and, in the latter case, belong to any quadrant of the real plane. Thus, we seek conditions that attribute these equilibrium points to the positive cone and describe their stability. The system depends on
a,
,
K, and
. To control the sign of
, we solve
in
, which yields
where
It will turn out that comparing
,
, and
yields a full description of the dynamics of the symmetric competition model. Consider the parameter plane
with
and
in the horizontal and vertical axes, respectively. We next show that the curve
as defined in (
48) and the bisector
divides
into four regions (for
,
). For this purpose, we need the following lemma.
Lemma 3. Consider Φ as defined in (48) along with , . Then, the following are true. - 1.
, so that Φ is strictly increasing.
- 2.
.
- 3.
Thus, .
- 4.
for any .
- 5.
for any .
- 6.
. That is, Φ crosses the bisector at , which entails that the crossing point is .
Proof. This follows from direct calculations. □
Lemma 3 implies that the curve
and the bisector
are the same at
. Furthermore, for
, the curve
increases (in
) faster than the bisector, and they cross at
, thus dividing the square
into the above-mentioned four regions
,
. See
Figure 6.
Theorem 9. Consider System (9) along with the symmetry conditions (45). Consider fixed values of K and a. Then, the following are true: - 1.
The equilibrium point is always in the positive cone and on the bisector of the positive cone.
- 2.
Global coexistence. For any such that , the semi-trivial equilibrium points and are unstable. The equilibrium is a global attractor to the positive cone.
- 3.
Priority effects. For any such that , the semi-trivial equilibrium points for are asymptotically stable and is unstable (a saddle). The stable manifold of determines the basins of attraction of the semi-trivial equilibrium points.
- 4.
Global bi-coexistence. For any such that , the semi-trivial equilibrium points and are unstable. There exist positive equilibrium points , , and in the positive cone. is unstable (a saddle). The latter two are asymptotically stable and located symmetrically with respect to the bisector. The stable manifold of determines the basins of attraction of and . See the bottom-left panel in Figure 7 and the right panel in Figure 8. - 5.
Coexistence or priority effects. For any such that , the semi-trivial equilibrium points , , and are locally asymptotically stable. There exist another two equilibrium points and in the positive cone, which are unstable (saddles) and located symmetrically with respect to the bisector. The stable manifolds of and determine the basins of attraction of , , and . See the bottom-right panel in Figure 7 and the left panel in Figure 8.
Proof. Statement 1 follows from the fact that
, as defined in (
46), and
.
As for statements 2 to 5, we recall that the stability of the semi-trivial equilibrium points is related to the balance between and ; see Theorem 6. Thus, for such that (that is, in ), there is global species coexistence, and, on the contrary, priority effects are possible if (that is, in ).
Indeed, note that the graph of
describes the combination of parameters such that
,
, and
collide, entailing a pitchfork bifurcation. That is to say that
gains/loses local stability as
and
lose/gain local stability. Direct calculations (meaning aided by Mathematica [
24]) showed that
and
exist and belong to the positive cone for values of
. Thus, outside these regions (i.e., in
and
),
is the only equilibrium point in the positive cone. With the system being competitive (
sensu [
20]), the stability of the semi-trivial equilibrium points yields statements 2 and 3.
Also, by applying the topological index (see Theorem 4.2, statement 2(a) in [
3]) along with a symmetry argument, the stability of the semi-trivial equilibrium points again yields statements 4 and 5. □
4.3. Numerical Simulations
Figure 8 displays two relevant bifurcation diagrams and phase portraits computed for the full model and not found under the settings presented up to now. The left panel displays simultaneously the coexistence or species 1 (or 2) wins regions along with the conditional coexistence (dark gray) region. Also, the right panel in
Figure 8 displays together the coexistence or species 2 wins region and the global bi-coexistence region (dark green, tiny region close to (1.5, 1.5) within the global coexistence region).
Figure 8.
Bifurcation diagrams, with
and
being the bifurcation parameters, and phase portraits with several orbits (in red). The color code is as in
Figure 2 and
Figure 6. Furthermore, the
left panel shows in dark gray the coexistence or one species extinction scenario that is illustrated by the corresponding phase portrait. The
right panel shows in dark green the bi-coexistence scenario along with the corresponding phase portrait. Parameter values of the phase portraits:
left,
,
,
,
;
right,
,
,
,
,
.
Figure 8.
Bifurcation diagrams, with
and
being the bifurcation parameters, and phase portraits with several orbits (in red). The color code is as in
Figure 2 and
Figure 6. Furthermore, the
left panel shows in dark gray the coexistence or one species extinction scenario that is illustrated by the corresponding phase portrait. The
right panel shows in dark green the bi-coexistence scenario along with the corresponding phase portrait. Parameter values of the phase portraits:
left,
,
,
,
;
right,
,
,
,
,
.