A Lotka-Voterra Competition Model
with Seasonal Succession
Sze-Bi Hsu
∗Department of Mathematics and
National Center for Theoretical Science
National Tsing Hua University
Hsinchu, Taiwan, R. O. China
E-mail: sbhsu@math.nthu.edu.tw
Xiao-Qiang Zhao
†Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL A1C 5S7, Canada
E-mail: zhao@mun.ca
Abstract A complete classification for the global dynamics of a Lotka-Voterra two species competition model with seasonal succession is obtained
∗Research supported in part by the National Council of Science of Republic of China. †Research supported in part by the NSERC of Canada and the MITACS of Canada.
via the stability analysis of equilibria and the monotone dynamical systems theory. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.
Key words and phrases: Lotka-Volterra model, seasonal succession, the period map, periodic solutions, global stability, saddle-point structure. 2010 MSC: 34C25, 37C65, 92D25
Short title for page headings: A seasonal succession model
1
Introduction
Season succession is an environmental variation on the growth of species due to the seasonal forcing in the nature. In temperate lakes there is a growing season for the phytoplankton and zooplankton during the warmer months af-ter which the species die off or form resting stages in the winaf-ter. It has been a fascinating subject for the plankton ecologist to study the modeling of non-equilibrium food dynamics by means of the season succession. The growth of the species is in fact driven by the combination of external forcing and inter-nal dynamics. Litchman and Klausmeimer [6] studied a mathematical model of the competition of two species for a single nutrient under fluctuating light with season succession in the chemostat. In [4], Klausmeimer applied the Flo-quet theory to several mathematical models including structured populations and food chains to obtain some analytic results. Klausmeier [5] also chose the well-known Rosenzweig-McArthur model as a prototypical predator-prey system to study the effect of season succession. In their numerical simu-lations, they found that in addition to the expected periodic oscilsimu-lations,
chaos may happen in some parameter range. In [9] and [3], there are some experimental data about the effect of season succession on the competition of phytoplankton species. However, there are few analytic results on these ecological models with season succession. Mathematically, the vector fields of these models are discontinuous and periodic in time t, but the associated period maps are continuous. In this paper, we choose two species competi-tion model of Lotka-Volterra type as a starting point to study analytically the possible effect of season succession. Indeed, this model is phenomenolog-ical and even conceptual, but we are able to give a complete mathematphenomenolog-ical classification for its global dynamics in terms of the involved parameters. From these analytical results, we further gain insights into effects of season succession on the competition outcomes.
The organization of the paper is as follows. In section 2, we first establish a threshold type result for the global dynamics of the logistic equation with season succession, and then for the discrete-time dynamical system generated by the period map of the two species competition model of Lotka-Volterra type with season succession, we prove the convergence of every forward orbit and the uniqueness of positive fixed point, and give sufficient conditions for the local stability and instability of the fixed points which are either on the boundaries or the interior of the first quadrant in the x–y plane. Section 3 is devoted to the study of the long-time behavior of solutions of our two species competition model. By employing two special orderings in R2 and the
monotone properties of the period map (see Lemma 2.1 (i) and (ii)), we obtain a complete classification for the global dynamics. More precisely, we give several sets of sufficient conditions for the global extinction of both species
(Theorem 3.1), the competitive exclusion (Theorem 3.2), the competitive coexistence of two species (Theorem 3.3), and the saddle-point structure (Theorem 3.4), respectively. It turns out that four possible outcomes of the classical Lotka-Volterra two species competition model are preserved by the period map of our competition model with season succession. In section 4, we discuss in detail the effects of two death rates in the bad season and the proportion of the good season on the competition outcomes.
2
Preliminaries
Let R+ := [0, ∞), Z+ be the set of all nonnegative integers, and ω > 0 be
given. We first consider the logistic equation with seasonal succession:
dx dt = −λx, mω ≤ t ≤ mω + (1 − φ)ω, dx dt = rx h 1 − x K i , mω + (1 − φ)ω ≤ t ≤ (m + 1)ω, (2.1) x(0) = x0 ∈ R+,
where m ∈ Z+, λ, r and K are positive constants, and φ ∈ (0, 1].
For any y0 ∈ R+, let y(t, y0) be the unique solution of the logistic equation
dy dt = ry h 1 − y K i , y(0, y0) = y0 ∈ R+. (2.2)
It then follows that the solution x(t, x0) can be determined uniquely as
fol-lows:
x(t, x0) = e−λtx(mω, x0), ∀t ∈ [mω, mω + (1 − φ)ω] ,
x(t, x0) = y (t − [mω + (1 − φ)ω], x(mω + (1 − φ)ω, x0)) ,
where m ∈ Z+. This implies that the solution x(t, x0) exists globally on
[0, ∞). We further have the following threshold dynamics for (2.1).
Theorem 2.1. (threshold dynamics) Let x(t, x0) be the unique solution
of system (2.1). Then the following two statements are valid:
(i) If rφ − λ(1 − φ) ≤ 0, then limt→∞x(t, x0) = 0 for all x0 ∈ R+.
(ii) If rφ − λ(1 − φ) > 0, then system (2.1) admits a unique positive ω-periodic solution x∗(t), and lim
t→∞(x(t, x0) − x∗(t)) = 0 for all x0 ∈
R+\ {0}.
Proof. It is easy to see that for any t ≥ 0, y(t, y0) is monotone increasing
in y0 ∈ R+. Note that the function f (y) := ry
£
1 − Ky¤ is strictly subho-mogeneous on R+ in the sense that f (αy) > αf (y), ∀y > 0, α ∈ (0, 1).
It then follows that for any t > 0, the map y(t, y0) is strictly
subhomo-geneous in y0 on R+. We consider the period map M associated with
system (2.1), that is, M(x0) = x(ω, x0), ∀x0 ∈ R+. It then follows that
M(x0) = y
¡
φω, e−λ(1−φ)ωx
0
¢
. Clearly, M(0) = 0, and M is monotone (in-creasing) on R+. By the differentiability of solutions with respect to initial
data, we easily see that
M0(0) = erφω· e−λ(1−φ)ω = e(rφ−λ(1−φ))ω.
Thus, [10, Theorem 2.3.4] implies that the period map M admits a threshold dynamics in terms of the sign of rφ − λ(1 − φ), and hence, statements (i) and (ii) are valid.
has the following explicit expression:
y(t, y0) =
Ky0
(K − y0)e−rt+ y0
, ∀t ≥ 0, y0 ∈ R+. (2.3)
Then the period map M of system (2.1) can be expressed as
M(x0) = Kx0e
−λ(1−φ)ω
(K − x0e−λ(1−φ)ω) e−rφω + x0e−λ(1−φ)ω
, ∀x0 ∈ R+.
A straightforward computation shows that in the case where rφ−λ(1−φ) > 0, the period map M has a unique positive fixed point
x∗ = K
¡
1 − eλ(1−φ)ω−rφω¢
1 − e−rφω .
Thus, one may directly employ the explicit expression of M(x0) to prove
Theorem 2.1, but our arguments supporting it also apply to other scalar evolution equations with seasonal succession.
Now we consider the Lotka-Volterra two species competition model with seasonal succession: dxi dt = −λixi, mω ≤ t ≤ mω + (1 − φ)ω, i = 1, 2, dx1 dt = r1x1 · 1 − x1 K1 ¸ − αx1x2, mω + (1 − φ)ω ≤ t ≤ (m + 1)ω, dx2 dt = r2x2 · 1 − x2 K2 ¸ − βx1x2, mω + (1 − φ)ω ≤ t ≤ (m + 1)ω, (x1(0), x2(0)) = x0 ∈ R2+, (2.4)
where m ∈ Z+, λi, ri, Ki, α and β are all positive constants, and φ ∈ (0, 1].
By a similar way of induction as in system (2.1), it follows that for any
x0 ∈ R2
+, system (2.4) admits a unique nonnegative global solution x(t, x0) on
the period map S on R2
+, that is, S(x0) = x(ω, x0), ∀x0 ∈ R2+. We define a
linear map L : R2 + by L(x1, x2) = ¡ e−λ1(1−φ)ωx 1, e−λ2(1−φ)ωx2 ¢ , ∀(x1, x2) ∈ R2+.
Let {Qt}t≥0 be solution semiflow associated with the Lotka-Volterra
compe-tition system: dx1 dt = r1x1 · 1 − x1 K1 ¸ − αx1x2, dx2 dt = r2x2 · 1 − x2 K2 ¸ − βx1x2, (2.5) (x1(0), x2(0)) = x0 ∈ R2+,
that is, Qt(x0), as a function of t, is the unique global solution of system
(2.5) on [0, ∞). It then easily follows that S(x0) = Q
φω(Lx0), ∀x0 ∈ R2+, i.e., S = Qφω◦ L. For x, y ∈ R2, we write x ≤ y if y − x ∈ R2 +; x < y if y − x ∈ R2+\ {0}; and x ¿ y if y − x ∈ Int(R2 +). Let K := {(x1, x2) ∈ R2 : x1 ≥ 0, x2 ≤ 0}.
Then we write x ≤K y if y − x ∈ K; x <K y if y − x ∈ K \ {0}; and x ¿K y
if y − x ∈ Int(K).
To describe the global dynamics of the discrete-time dynamical system
{Sn}
n≥0, we need a series of lemmas.
Lemma 2.1. (monotonicity and convergence) The map S has the
following properties on R2 +:
(i) If x ≤K y, then S(x) ≤K S(y).
(iii) For any x0 ∈ R2
+, the sequence of points Sn(x0) converges to a fixed
point of S as n → ∞.
Proof. By the comparison theorem of competitive systems (see, e.g., [8,
The-orem B.4]), it follows that Qt(x0) ≤K Qt(y0), ∀t ≥ 0, x0, y0 ∈ R2+ with
x0 ≤
K y0, and hence, the property (i) holds for the map S. For any
given t0 ≥ 0, and x, y ∈ R2+ with Qt0(x) ≤ Qt0(y), it is easy to see that
u(t) := Qt0−t(x) and v(t) := Qt0−t(y) are two solutions of a two dimensional
cooperative system for t ∈ [0, t0]. Since u(0) = Qt0(x) ≤ Qt0(y) = v(0),
the comparison theorem of cooperative systems (see, e.g., [8, Corollary B.2]) implies that x = u(t0) ≤ v(t0) = y. By this property of the semiflow {Qt}t≥0
and the expression of the linear map L, it then follows that the property (ii) holds for the map S. Now the property (iii) is a consequence of (i), (ii), and the same arguments as in [8, Theorem 7.4.2] (see also the proof [1, Theorem 4.1]).
Lemma 2.2. (stability of semitrivial fixed points) Assume that
r1φ − λ1(1 − φ) > 0 and let x∗1(t) be the unique positive ω-periodic solution
of system (2.1) with r = r1 and K = K1. Let x∗ = (x∗1(0), 0), DS(x∗) be the
Jacobian matrix (i.e., the Frech´et derivative) of S at x∗, and r(DS(x∗)) be
the spectral radius of the linear operator DS(x∗). Then following statements
are valid:
(i) If r2φ − λ2(1 − φ) > βKr11 (r1φ − λ1(1 − φ)), then r(DS(x∗)) > 1, and
hence, x∗ is an unstable fixed point of S.
(ii) If r2φ − λ2(1 − φ) < βKr11 (r1φ − λ1(1 − φ)), then r(DS(x∗)) < 1, and
An analogous stability result also holds for the case where r2φ−λ2(1−φ) > 0.
Proof. Let f (x) be the right-hand side vector field of system (2.5). Then we
have Df (x) = r1−2rK11x1− αx2 −αx1 −βx2 r2− 2rK22x2− βx1 . (2.6)
Let u(t, x) := Qt(x), and V (t, x) := Dxu(t, x). It then follows that for any
given x ∈ R2
+, the matrix function V (t, x) satisfies
dV (t)
dt = Df (u(t, x))V (t), V (0) = I.
Since S = Qφω ◦ L, the chain rule implies that
DS(x∗) = DQ φω(Lx∗) e−λ1(1−φ)ω 0 0 e−λ2(1−φ)ω .
Thus, DQφω(Lx∗) = Dxu(φω, Lx∗) = V (φω, Lx∗). Let x∗ = (x∗1, 0). Then
Lx∗ = ¡e−λ1(1−φ)ωx∗
1, 0
¢
. By the expression (2.3), we have u(t, Lx∗) =
(u∗ 1(t), 0) with u∗1(t) = K1x ∗ 1e−λ1(1−φ)ω (K1− x∗1e−λ1(1−φ)ω) e−r1t+ x∗1e−λ1(1−φ)ω , (2.7) and hence, Df (u(t, Lx∗)) = r1−2rK11u∗1(t) −αu∗1(t) 0 r2− βu∗1(t) . It then follows that
V (φω, Lx∗) = e Rφω 0 ³ r1−2r1K1u∗1(t) ´ dt ∗ 0 eR0φω(r2−βu∗1(t))dt .
Thus, the matrix DS(x∗) has two positive eigenvalues µ 1 and µ2 given by µ1 = e Rφω 0 ³ r1−2r1K1u∗1(t) ´ dt · e−λ1(1−φ)ω, µ 2 = e Rφω 0 (r2−βu∗1(t))dt· e−λ2(1−φ)ω.
By the formula (2.7) and elementary computations, or by integrating the logistic equation for u∗
1(t) (see, e.g., the proof of Lemma 2.4 (i)), it then
follows that Z φω 0 u∗1(t)dt = K1 r1 (r1φ − λ1(1 − φ)) ω.
A straightforward calculation further shows that
µ1 = e−(r1φ−λ1(1−φ))ω < 1, µ2 = e ³
r2φ−λ2(1−φ)−βK1r1 (r1φ−λ1(1−φ))
´
ω,
which implies that the statements (i) and (ii) hold.
Lemma 2.3. (stability of positive fixed point) Let ¯x be a positive fixed point of S, and r(DS(¯x)) be the spectral radius of the linear operator DS(¯x). Then following statements are valid:
(i) If r1r2
K1K2 < αβ, then r(DS(¯x)) > 1, and hence, ¯x is an unstable fixed
point of S. (ii) If r1r2
K1K2 > αβ, then r(DS(¯x)) < 1, and hence, ¯x is an asymptotically
stable fixed point of S.
Proof. Let f (x), u(t, x) and V (t, x) be defined as in the proof of Lemma 2.2.
It then follows that
DS(¯x) = V (φω, L¯x) e−λ1(1−φ)ω 0 0 e−λ2(1−φ)ω .
Let ¯u(t) = (¯u1(t), ¯u2(t)) := Qt(L¯x) and V (t) := V (t, L¯x). Then dV (t) dt = Df (¯u(t))V (t), V (0) = I. Let P (t) := ¯u11(t) 0 0 1 ¯ u2(t) .
By the expression of Df (x) in (2.6) and a straightforward computation, it then follows that under a change of variable w = P (t)v, the linear equation
dv
dt = Df (¯u(t))v becomes the following one:
dw dt = −Kr11u¯1(t) −α¯u2(t) −β ¯u1(t) −Kr22u¯2(t) w := A(t)w. (2.8)
Let W (t) be the monodromy matrix of system (2.8), that is, W (t) satisfies
dW (t)
dt = A(t)W (t), W (0) = I.
Thus, we have W (t) = P (t)V (t)P−1(0), and hence, V (t) = P−1(t)W (t)P (0), ∀t ≥
0. Let ¯x = (¯x1, ¯x2). Note that
¯
u(0) = L¯x =¡e−λ1(1−φ)ωx¯
1, e−λ2(1−φ)ωx¯2
¢
,
and ¯u(φω) = Qφω(L¯x) = S(¯x) = ¯x. It then follows that
P (0) = e λ1(1−φ)ω ¯ x1 0 0 eλ2(1−φ)ω ¯ x2 , P (φω) = x¯11 0 0 1 ¯ x2 ,
and hence, DS(¯x) = V (φω) e−λ1(1−φ)ω 0 0 e−λ2(1−φ)ω = P−1(φω)W (φω)P (0) e−λ1(1−φ)ω 0 0 e−λ2(1−φ)ω = x¯1 0 0 x¯2 W (φω) x¯1 0 0 x¯2 −1 ,
which implies that DS(¯x) is similar to W (φω). Thus, we have r(DS(¯x)) = r(W (φω)). Let Z(t) = 1 0 0 −1 W (t) 1 0 0 −1 −1 . Then Z(t) satisfies dZ dt = −Kr11u¯1(t) α¯u2(t) β ¯u1(t) −Kr22u¯2(t) Z := B(t)Z, Z(0) = I. (2.9) Since the matrix B(t) is cooperative and irreducible, it follows from [8, Theo-rem B.3] that for each t > 0, every element of Z(t) is positive. Clearly, W (φω) is similar to Z(φω), and hence, r(W (φω)) = r(Z(φω)). By Perron-Frobenius theorem (see, e.g., [8, Theorem A.4]), ρ2 := r(Z(φω)) is a simple eigenvalue
of Z(φω) with a positive eigenvector e = (e1, e2)T. Let ρ1 be another
eigen-value of Z(φω). By Liouville’s formula, we have 0 < ρ1ρ2 = detZ(φω) =
Z(t)e. Then z(φω) = Z(φω)e = ρ2e, and zi(t) > 0, ∀t ≥ 0, i = 1, 2. Since dz1(t) dt = − r1 K1 ¯ u1(t)z1(t) + α¯u2(t)z2(t), dz2(t) dt = β ¯u1(t)z1(t) − r2 K2 ¯ u2(t)z2(t), it follows that βdz1(t) dt + r1 K1 dz2(t) dt = µ αβ − r1r2 K1K2 ¶ ¯ u2(t)z2(t), ∀t ≥ 0.
Integrating the above equation for t from 0 to φω, we then obtain µ βe1+ r1 K1 e2 ¶ (ρ2− 1) = µ αβ − r1r2 K1K2 ¶ Z φω 0 ¯ u2(t)z2(t)dt. Since ³ βe1+Kr11e2 ´
> 0 and R0φωu¯2(t)z2(t)dt > 0, we see that ρ2 − 1 has
the same sign as αβ − r1r2
K1K2. This implies that the statements (i) and (ii)
hold.
For the sake of convenience, we introduce the following notations:
A := Kr11 α β r2 K2 , B := (r1φ − λ1(1 − φ)) ω (r2φ − λ2(1 − φ)) ω . Then we have the following observation.
Lemma 2.4. (uniqueness of positive fixed point) The following two
statements are valid:
(i) If S has a positive fixed point ¯x, then y := R0φωQt(L¯x)dt is a positive
solution of the linear algebraic system Ay = B. (ii) If r1r2
Proof. Let x(t) = (x1(t), x2(t)) := Qt(L¯x) and yi :=
Rφω
0 xi(t)dt, i = 1, 2.
Then x(φω) = Qφω(L¯x) = S(¯x) = ¯x, and xi(t) > 0, ∀t ≥ 0, i = 1, 2. Since
(x1(t), x2(t)) satisfies equation (2.5), it follows that
r1 · 1 − x1(t) K1 ¸ − αx2(t) = x0 1(t) x1(t) , r2 · 1 − x2(t) K2 ¸ − βx1(t) = x0 2(t) x2(t) .
Integrating the above two equations for t from 0 to φω, we then obtain
r1φω − r1 K1 y1− αy2 = ln ¯x1− ln ¡ e−λ1(1−φ)ωx¯ 1 ¢ = λ1(1 − φ)ω, r2φω − r2 K2 y2− βy1 = ln ¯x2− ln ¡ e−λ2(1−φ)ωx¯ 2 ¢ = λ2(1 − φ)ω.
This implies that the statement (i) holds.
Now we prove the uniqueness of the positive fixed point of S. Suppose, by contradiction, that S has two distinct positive fixed points ¯x and ¯y. Since detA = r1r2
K1K2 − αβ 6= 0, the linear system Ay = B has a unique solution. By
the statement (i), it then follows that Z φω 0 Qt(L¯x)dt = Z φω 0 Qt(L¯y)dt. (2.10)
Since any two points in R2 are ordered with respect to one of two orderings
≤ and ≤K, we can assume, without loss of generality, that either ¯y < ¯x, or
¯
y <K x. In the case where ¯¯ y < ¯x, we have Qφω(L¯y) = ¯y ≤ ¯x = Qφω(L¯x).
By the comparison theorem of two dimensional competition systems in the negative direction (see, e.g., the proof of Lemma 2.1 (ii)), it then follows that
Qt(L¯y) ≤ Qt(L¯x), ∀t ∈ [0, φω]. Since L¯y < L¯x, we see that Qt(L¯y) 6≡ Qt(L¯x)
on [0, φω]. Thus, we have R0φωQt(L¯y)dt <
Rφω
(2.10). In the case where ¯y <K x, we have L¯¯ y <K L¯x. By the comparison
the-orem of two dimensional competition systems (see, e.g., [8, Thethe-orem B.4]), it then follows that Qt(L¯y) ≤K Qt(L¯x), ∀t ∈ [0, φω]. Since L¯y <K L¯x, we have
Qt(L¯y) 6≡ Qt(L¯x) on [0, φω], and hence,
Rφω
0 Qt(L¯y)dt <K
RφT
0 Qt(L¯x)dt.
This also contradicts (2.10).
3
Global dynamics
Throughout this section, we continue to use the notations introduced in section 2. In particular, x(t, x0) denotes the unique solution of system (2.4),
S = Qφω◦ L is the period map associated with system (2.4), and x∗i(t) is the
unique positive ω-periodic solution of system (2.1) with r = ri and K = Ki
whenever riφ − λi(1 − φ) > 0.
For the sake of convenience, we also use the notations 0 = (0, 0), C1 :=
{(x1, 0) : x1 ∈ R+}, and C2 := {(0, x2) : x2 ∈ R+}. A vector function
f (t) is said to be asymptotic to an ω-periodic function g(t) provided that
limt→∞|f (t) − g(t)| = 0.
Theorem 3.1. (global extinction) The following statements are valid:
(i) If riφ − λi(1 − φ) < 0, ∀i = 1, 2, then the zero solution is globally
asymptotically stable for system (2.4) in R2 +.
(ii) If r1φ − λ1(1 − φ) > 0 and r2φ − λ2(1 − φ) < 0, then any solution of
system (2.4) in R2
+\ C2 is asymptotic to (x∗1(t), 0).
(iii) If r1φ − λ1(1 − φ) < 0 and r2φ − λ2(1 − φ) > 0, then any solution of
system (2.4) in R2
Proof. In three cases (i), (ii) and (iii), at least one component of the vector B is negative, and hence, Ax = B has no positive solution. By Lemma 2.4
(i), it then follows that the map S has no positive fixed point.
In case (i), Theorem 2.1 (i) implies that S has no nonzero fixed point on the boundary of R2
+. By Lemma 2.1 (iii), it follows that every forward orbit
of Sn in R2
+ converges to 0. As in the proof of Lemma 2.2, it is easy to see
that DS(0) = DQφω(0) e−λ1(1−φ))ω 0 0 e−λ2(1−φ)ω = e(r1φ−λ1(1−φ))ω 0 0 e(r2φ−λ2(1−φ))ω .
Thus, we see that r(DS(0)) < 1, and hence, 0 is a linearly stable fixed point of S. Consequently, we obtain the global asymptotic stability of the zero solution for the periodic system (2.4).
In case (ii), Theorem 2.1 (i) implies that S has exactly one nonzero fixed point (x∗
1(0), 0) on the boundary of R2+. Note that DS(0) has two positive
eigenvalues: one is greater than 1 and the other is less than 1. Thus, 0 is a saddle fixed point of S. By Lemma 2.1 (iii), it then follows that every forward orbit for S in R2
+\ C2 converges to (x∗1(0), 0). This implies that the
statement (ii) is valid. By symmetric arguments, we see that the statement (iii) also holds.
Theorem 3.2. (competitive exclusion) Assume that riφ − λi(1 − φ) >
0, ∀i = 1, 2. Then the following statements are valid:
αK2
r2 (r2φ − λ2(1 − φ)), then the solution (x
∗
1(t), 0) is globally
asymptot-ically stable for system (2.4) in R2 +\ C2.
(ii) If r2φ − λ2(1 − φ) > βKr11 (r1φ − λ1(1 − φ)) and r1φ − λ1(1 − φ) <
αK2
r2 (r2φ − λ2(1 − φ)), then the solution (0, x
∗
2(t)) is globally
asymptot-ically stable for system (2.4) in R2 +\ C1.
Proof. We consider two lines on y1-y2 plane
l1 : r1 K1 y1+ αy2 = (r1φ − λ1(1 − φ)) ω > 0, and l2 : βy1+ r2 K2 y2 = (r2φ − λ2(1 − φ)) ω > 0.
It is easy to verify that the y1 and y2 intercepts of l1 are greater than those
of l2 in case (i); and the y1 and y2 intercepts of l2 are greater than those of
l1 in case (ii). This implies that both l1 and l2 do not intersect in R2+. Then
Lemma 2.4 (i) implies that S has no positive fixed point. By the expression of DS(0) as in the proof of Theorem 3.1, it follows that 0 is a unstable fixed point of S in both x1 and x2 directions. From Lemma 2.2 and its proof, we
further see that in case (i), (x∗
1(0), 0) is an asymptotic stable fixed point of
S and (0, x∗
2(0)) is a unstable fixed point of the saddle type. By Lemma 2.1
(iii), it then follows that (x∗
1(0), 0) globally asymptotically stable for S in
R2
+\ C2, which implies the statement (i). The statement (ii) can be proved
in a similar way.
Theorem 3.3. (competitive coexistence) Assume that riφ−λi(1−φ) >
0, ∀i = 1, 2. If r2φ − λ2(1 − φ) > βKr11 (r1φ − λ1(1 − φ)) and r1φ − λ1(1 − φ) >
αK2
solution ¯x(t), and ¯x(t) is globally asymptotically stable for system (2.4) in Int(R2
+).
Proof. As in the proof of Theorem 3.2, 0 is a unstable fixed point of S in
both x1 and x2 directions. By Lemma 2.2 and its proof, it follows that both
(x∗
1(0), 0) and (0, x∗2(0)) are two unstable fixed points of the saddle type.
Thus, Lemma 2.1 (iii) implies that for any x0 ∈ Int(R2
+), Sn(x0) converges
to a fixed point of S in Int(R2
+) as n → ∞. Since r2φ − λ2(1 − φ) > 0 and r2φ − λ2(1 − φ) > βK1 r1 (r1φ − λ1(1 − φ)) > βK1 r1 ·αK2 r2 (r2φ − λ2(1 − φ)) , we then have r1r2
K1K2 > αβ. In view of Lemma 2.4 (ii), S has a unique positive
fixed point ¯x, and hence, ¯x is globally attractive for S in Int(R2
+). Further,
Lemma 2.3 (ii) implies the local stability of the fixed point ¯x. Consequently,
we have the above stated conclusion for solutions of system (2.4).
Theorem 3.4. (saddle-point structure) Assume that riφ−λi(1−φ) >
0, ∀i = 1, 2. If r2φ − λ2(1 − φ) < βKr11 (r1φ − λ1(1 − φ)) and r1φ − λ1(1 − φ) <
αK2
r2 (r2φ − λ2(1 − φ)), then system (2.4) admits a unique positive ω-periodic
solution ¯x(t), and there exists a continuous, unbounded and one-dimensional curve Γ ⊂ R2
+such that both 0 and ¯x(0) are in Γ, and the following statements
are valid:
(a) If x0 ∈ Γ \ {0}, then the solution x(t, x0) is asymptotic to ¯x(t).
(b) If x0 >
K y0 for some y0 ∈ Γ, then the solution x(t, x0) is asymptotic
to (x∗
(c) If x0 <
K z0 for some z0 ∈ Γ, then the solution x(t, x0) is asymptotic to
(0, x∗
2(t)).
Proof. In view of Lemma 2.2, both (x∗
1(0), 0) and (0, x∗2(0)) are asymptotic
stable fixed points of S. Let B1 and B2 be the basins of attraction of (x∗1(0), 0)
and (0, x∗
2(0)) for S in R2+, respectively. It then follows that B1 and B2 are
relatively open in R2
+. Now we claim that S has at least one fixed point ¯x
in Int(R2
+). Otherwise, Lemma 2.1 (iii) implies that R2+ \ {0} = B1 ∪ B2,
which contradicts the connectedness of R2
+\ {0}. Note that the existence
of the positive fixed point ¯x is also a straightforward consequence of [2,
Proposition 2.1] on connecting orbits for abstract competitive systems. Since
r2φ − λ2(1 − φ) > 0 and r2φ − λ2(1 − φ) < βK1 r1 (r1φ − λ1(1 − φ)) < βK1 r1 ·αK2 r2 (r2φ − λ2(1 − φ)) , we then have r1r2
K1K2 < αβ. Thus, Lemma 2.4 (ii) implies that ¯x is a unique
positive fixed point of S. Further, we see from Lemma 2.3 and its proof that
DS(¯x) has two positive eigenvalues ρ1 and ρ2 such that ρ1ρ2 < 1 and ρ2 > 1.
Since ρ1 = (ρ1ρ2) · ρ12 < 1, the fixed point ¯x is of the saddle point type.
It then follows that there exists an open disc U centered at ¯x such that U
contains a one-dimensional and smooth stable manifold Ws
loc(¯x) at ¯x.
Define Γ := R2
+\ (B1∪ B2). It is easy to see that 0 ∈ Γ, Wlocs (¯x) ⊂ Γ,
and S(Γ) = Γ. Further, Lemma 2.1 (iii) implies that limn→∞Snx = ¯x, ∀x ∈
Γ \ {0}. We proceed with the following three claims on Γ.
Claim 1. Any two distinct points in Γ are strongly ordered with respect to
Note that if one of these two points is 0, then the claim is obviously true due to the fact that Γ \ {0} ⊂ Int(R2
+). It then suffices to show that any
two distinct points x and y in Γ \ {0} are not related by <K. Assume, by
contradiction, that there are two points x, y ∈ Γ\{0} such that x <K y. Then
Snx ¿
K Sny, ∀n ≥ 1. By Lemma 2.1 (iii), it follows that limn→∞Snx = ¯x =
limn→∞Sny, and hence, there exists n0 ≥ 1 such that Snx, Sny ∈ U, ∀n ≥ n0.
For any given z ∈ [[Sn0x, Sn0y]]
K := {w ∈ R2 : Sn0x ¿K w ¿K Sn0y} ⊂ U,
we have Sn+n0x ¿
K Snz ¿K Sn+n0y, ∀n ≥ 0, and hence, limn→∞Snz = ¯x.
This implies that the open set [[Sn0x, Sn0y]]
K ⊂ Wlocs (¯x), which contradicts
the fact that dim Ws
loc(¯x) = 1. This proves the above Claim 1.
Claim 2. For any x, y ∈ Γ with x ¿ y, we have Sx ¿ Sy and S−1x ¿ S−1y.
Indeed, for any x, y ∈ Γ with x ¿ y, we have Sx 6= Sy, and Sx, Sy ∈ Γ. Then Claim 1 implies that either Sx ¿ Sy, or Sx À Sy. If Sx À Sy, then Lemma 2.1 implies that x ≥ y, which is impossible. Thus, we must have
Sx ¿ Sy. Since S−1x 6= S−1y, and S−1x, S−1y ∈ Γ, Claim 1 implies that
either S−1x ¿ S−1y, or S−1x À S−1y. If S−1x À S−1y, then it follows
from what we have just proved that x = S(S−1x) À S(S−1y) = y, which is
impossible. Therefore, we have S−1x ¿ S−1y.
Claim 3. For any y ∈ Γ \ {0}, there exist x, z ∈ Γ \ {0} such that x ¿ y ¿ z.
Assume, by contradiction, that there exists some y ∈ Γ \ {0} such that ¡
y + Int(R2 +)
¢
∩ (Γ \ {0}) = ∅. Choose two points u, v ∈ y + R2
+ such that
v ¿K u and v ≤K y ≤K u, and let l be the closed segment connecting u
and v. Then we have l ∩ Γ = ∅. Further, the above Claim 1 implies that that u, v 6∈ Γ. Since limn→∞Sny = ¯x and Snv ≤K Sny ≤K Snu, ∀n ≥ 1, it
l ∩ Γ = ∅ and both (x∗
1(0), 0) and (0, x∗2(0)) are locally asymptotically stable.
This, together with Lemma 2.1 (iii), implies that for any point x0 ∈ l, there
exists an open neighborhood U0 of x0 in R2
+ such that limn→∞Sn(y0) =
limn→∞Sn(x0), ∀y0 ∈ U0. By the compactness of l in R2+, we then have
limn→∞Snu = limn→∞Snv, which contradicts the fact that (0, x∗2(0)) ¿K
(x∗
1(0), 0). This shows that for any y ∈ Γ \ {0}, there exists some z ∈ Γ \ {0}
such that y ¿ z. The other part of Claim 3 can be proved in a similar way. Now we prove that any point y ∈ Γ \ {0} lies in some continuous, open and one-dimensional arc contained in Γ. Let y ∈ Γ \ {0} be given. By Claim 3, there exist x, z ∈ Γ \ {0} such that x ¿ y ¿ z. Thus, Claim 2 implies that Snx ¿ Sny ¿ Snz, ∀n ≥ 1. Since lim
n→∞Snx = ¯x = limn→∞Snz,
there exists an integer n0 such that
[Snx, Snz] := {u ∈ R2+ : Snx ≤ u ≤ Snz} ⊂ U, ∀n ≥ n0.
Define
Γ0 := [[Sn0x, Sn0z]]Γ= {v ∈ Γ : Sn0x ¿ v ¿ Sn0z}.
It then follows that Γ0 ⊂ Wlocs (¯x), and hence, Γ0 is a continuous,
one-dimensional, and strongly ordered arc. Thus, Claim 2 implies that S−n0(Γ
0) =
(S−1)n0
(Γ0) is also a continuous, one-dimensional, and strongly ordered arc.
It easily follows from Claim 2 that y ∈ [[x, z]]Γ= S−n0(Γ0).
Let x ∈ R2
+ be given such that x >K y for some y ∈ Γ. Then x 6∈ Γ and
Snx ≥
K Sny, ∀n ≥ 1. Since limn→∞Sny = ¯x, it follows from Lemma 2.1
(iii) that limn→∞Snx = (x∗1(0), 0) ≥K x. By similar arguments, we can show¯
that limn→∞Snx = (0, x∗2(0)) ≤K x for any given x ∈ R¯ 2+ with x <K z for
some z ∈ Γ. Consequently, we have the above stated conclusion for solutions of system (2.4).
4
Discussions
Consider the following classical Lotka-Volterra two species competition model:
dx1 dt = r1x1 · 1 − x1 K1 ¸ − αx1x2, dx2 dt = r2x2 · 1 − x2 K2 ¸ − βx1x2, (4.1) (x1(0), x2(0)) = x0 ∈ R2+.
It is well-known that there are four cases of the competition outcome in model (4.1) (see, e.g., [7, Section 3.5]), namely:
A. If K1 > rβ2 and K2 < rα1, then species 1 wins the competition.
B. If K1 < rβ2 and K2 > rα1, then species 2 wins the competition.
C. If K1 < rβ2 and K2 < rα1, then two species coexist at a stable equilibrium.
D. If K1 > rβ2 and K2 > rα1, then the bistability occurs for two species in
the sense that there exists a separatrix curve Γ connecting (0, 0) to the infinity in R2
+ such that species 2 wins whenever the initial distribution
is above Γ, while species 1 wins whenever the initial distribution is below Γ.
In this paper, we studied the Lotka-Volterra two species competition model with seasonal succession (2.4). The results can be summarized according to five cases, namely:
E. Two species go to extinction. I. Species 1 wins the competition.
II. Species 2 wins the competition.
III. Two species coexist at a stable periodic state. IV. Bistability occurs for two species.
From Theorem 3.1 (i), we first see that if
riφ − λi(1 − φ) < 0, ∀i = 1, 2, (4.2)
then the outcome E occurs. By Theorem 3.1 (ii) and Theorem 3.2 (i), it follows that the outcome I occurs provided either
r1φ − λ1(1 − φ) > 0, r2φ − λ2(1 − φ) < 0, (4.3) or riφ − λi(1 − φ) > 0, ∀i = 1, 2, r2φ − λ2(1 − φ) < βK1 r1 (r1φ − λ1(1 − φ)) , (4.4) r1φ − λ1(1 − φ) > αK2 r2 (r2φ − λ2(1 − φ)) .
In view of Theorem 3.1 (iii) and Theorem 3.2 (ii), the outcome II occurs provided either r1φ − λ1(1 − φ) < 0, r2φ − λ2(1 − φ) > 0, (4.5) or riφ − λi(1 − φ) > 0, ∀i = 1, 2, r2φ − λ2(1 − φ) > βK1 r1 (r1φ − λ1(1 − φ)) , (4.6) r1φ − λ1(1 − φ) < αK2 r2 (r2φ − λ2(1 − φ)) .
Theorem 3.3 implies that the outcome III occurs provided riφ − λi(1 − φ) > 0, ∀i = 1, 2, r2φ − λ2(1 − φ) > βK1 r1 (r1φ − λ1(1 − φ)) , (4.7) r1φ − λ1(1 − φ) > αK2 r2 (r2φ − λ2(1 − φ)) .
Finally, we see from Theorem 3.4 that the outcome IV occurs provided
riφ − λi(1 − φ) > 0, ∀i = 1, 2, r2φ − λ2(1 − φ) < βK1 r1 (r1φ − λ1(1 − φ)) , (4.8) r1φ − λ1(1 − φ) < αK2 r2 (r2φ − λ2(1 − φ)) .
There are four parameters λ1,λ2, φ and ω related to the season succession.
Our analytic results show that the period ω is independent of any outcome in the effect of season succession. Theorem 3.1 (i)-(iii) states that if the death rate λi of the i-th species in the bad season is too large, i.e., λi > 1−φriφ, then
the i-th species becomes extinct as time becomes large. Thus, in the rest of this section, we always assume that 0 < λi < 1−φriφ, ∀i = 1, 2.
4.1
The effect of λ
1and λ
2In this subsection, we fix all parameters except λ1 and λ2, and let L1 and L2
represent two lines in the λ1–λ2 plane with their equations given, respectively,
by L1 : r2φ − λ2(1 − φ) = βK1 r1 (r1φ − λ1(1 − φ)) , and L2 : r1φ − λ1(1 − φ) = αK2 r2 (r2φ − λ2(1 − φ)) .
Case A: r1 > αK2 and r2 < βK1. In this case, if φ = 1, i.e., there is no
bad season, then species 1 wins the competition. Let 0 < φ < 1 be given, and define P1 := µ r1φ 1 − φ, 0 ¶ , P2 := µ 0, r2φ 1 − φ ¶ , R1 := µ r1φ 1 − φ · 1 − r2 βK1 ¸ , 0 ¶ , and R2 := µ r1φ 1 − φ · 1 − αK2 r1 ¸ , 0 ¶ .
Then we have two subcases.
Subcase A1: K1K2 > rαβ1r2. In some parameter regions in the λ1–λ2 plane,
we may have bistability case, or the competition outcome may be reversed to the case where species 2 wins the competition, see Figure 1.
6 -λ2 λ1 P1 P2 %% %% %% %% %% %% ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ • • R2 R1 L2 L1 I IV II Figure 1: Subcase A1
we may have the stable coexistence case, or the competition outcome may be reversed to the case where species 2 wins the competition, see Figure 2.
6 -λ2 λ1 P1 P2 %% %% %% %% %% %% ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ • • R1 R2 L1 L2 I III II Figure 2: Subcase A2
Case B: r1 < αK2 and r2 > βK1. In this case, if φ = 1, i.e., there is no
bad season, then the species 2 wins the competition. Let 0 < φ < 1 be given, and define Q1 := µ 0, r2φ 1 − φ · 1 − βK1 r2 ¸¶ , and Q2 := µ 0, r2φ 1 − φ · 1 − r1 αK2 ¸¶ .
Then we have two subcases.
Subcase B1: K1K2 > rαβ1r2. In some parameter regions in the λ1–λ2 plane,
the bistability may occur, or the competition outcome may be reversed to the case where the species 1 wins the competition, see Figure 3.
6 -λ2 λ1 P1 P2 ,, ,, ,, ,, ,, ,, !!!! !!!! !!!! • • Q1 Q2 L1 L2 I IV II Figure 3: Subcase B1
Subcase B2: K1K2 < rαβ1r2. In some parameter regions in the λ1–λ2 plane,
we may have the stable coexistence for two species, or the competition out-come may be reversed to the case where the species 1 wins the competition, see Figure 4.
Case C: r1 > αK2and r2 > βK1. In this case, if φ = 1, i.e., there is no bad
season, then we have the stable coexistence for two species. For φ ∈ (0, 1), the competition outcome may be reversed to the case of the competitive exclusion, see Figure 5.
Case D: r1 < αK2 and r2 < βK1. In this case, if φ = 1, i.e., there is no
bad season, then we have the bistability case. For φ ∈ (0, 1), the competition outcomes may be reversed to the case of the competitive exclusion, see Figure 6.
6 -λ2 λ1 P1 P2 ,, ,, ,, ,, ,, ,, !!!! !!!! !!!! • • Q2 Q1 L2 L1 I III II Figure 4: Subcase B2
4.2
The effect of φ
In this subsection, we fix all parameters except φ ∈ (0, 1]. Define two func-tions f (φ) and g(φ) by f (φ) := (r2φ − λ2(1 − φ)) − βK1 r1 (r1φ − λ1(1 − φ)) , and g(φ) := (r1φ − λ1(1 − φ)) − αK2 r2 (r2φ − λ2(1 − φ)) .
Without loss of generality, we assume that
λ2
r2 + λ2
≤ λ1 r1+ λ1
. (4.9)
This is because the analogous conclusions can be obtained for the reversed case of (4.9) by an exchange of the positions of two competing species x1 and
6 -λ2 λ1 P1 P2 ´´ ´´ ´´ ´´ ´´ ´´ • Q1 L2 • R2 L1 I III II Figure 5: Case C
Biologically, this means that the death rate λ2 of species 2 is smaller than λ1
in comparison with intrinsic growth rates.
We note that riφ − λi(1 − φ) > 0 if and only φ > riλ+λi i, i = 1, 2. It is easy
to verify that the following statements are valid: (i) If r2 > βK1, then f (φ) > 0 for all φ ∈
h
λ1
r1+λ1, 1
i . (ii) If r2 < βK1, then f (φ) > 0 for all φ ∈
h λ1 r1+λ1, φ ∗ f ´
, and f (φ) < 0 for all
φ ∈¡φ∗ f, 1 ¤ , where f (φ∗ f) = 0 and φ∗ f = ³ βK1λ1 r1 − λ2 ´ (βK1− r2) + ³ βK1λ1 r1 − λ2 ´.
(iii) If r1 > αK2, then g(φ) < 0 for all φ ∈
h λ1 r1+λ1, φ ∗ g ´
6 -λ2 λ1 P1 P2 ´´ ´´ ´´ ´´ ´´ ´´ • Q2 L1 • R1 L2 I IV II Figure 6: Case D φ ∈ (φ∗ g, 1], where g(φ∗g) = 0 and φ∗g = ³ λ1− αKr22λ2 ´ (r1− αK2) + ³ λ1− αKr22λ2 ´.
(iv) If r1 < αK2, then g(φ) < 0 for all φ ∈
h
λ1
r1+λ1, 1
i .
Case A: r1 > αK2 and r2 < βK1. In this case, if φ = 1, i.e., there is
no bad season, then species 1 wins the competition. It is easy to verify that
φ∗
g < φ∗f if and only if αβK1K2 < r1r2. Thus, we have two subcases.
Subcase A1: K1K2 > rαβ1r2. It then follows that the following statements
are valid: (1) If φ ∈ ³ λ1 r1+λ1, φ ∗ f ´
, then f (φ) > 0 and g(φ) < 0, and hence, species 2 wins the competition.
(2) If φ ∈ (φ∗
f, φ∗g), then f (φ) < 0 and g(φ) < 0, and hence, the bistability
occurs for two species. (3) If φ ∈ (φ∗
g, 1), then f (φ) < 0 and g(φ) > 0, and hence, species 1 wins
the competition.
Subcase A2: K1K2 < rαβ1r2. It then follows that the following statements
are valid: (1) If φ ∈ ³ λ1 r1+λ1, φ ∗ g ´
, then g(φ) < 0 and f (φ) > 0, and hence, species 2 wins the competition.
(2) If φ ∈ (φ∗
g, φ∗f), then g(φ) > 0 and f (φ) > 0, and hence, two species
coexist at a stable equilibrium. (3) If φ ∈ (φ∗
f, 1), then g(φ) > 0 and f (φ) < 0, and hence, species 1 wins
the competition.
Case B: r1 < αK2 and r2 > βK1. In this case, if φ = 1, i.e., there is no
bad season, then the species 2 wins the competition. It is easy to verify that
g(φ) < 0 and f (φ) > 0 for all φ ∈
h
λ1
r1+λ1, 1
i
. Thus, species 2 still wins the competition.
Case C: r1 > αK2 and r2 > βK1. In this case, if φ = 1, i.e., there is no
bad season, then we have the stable coexistence. Given φ ∈ (0, 1), we have
f (φ) > 0; and g(φ) < 0 whenever φ ∈ ³ λ1 r1+λ1, φ ∗ g ´ , and g(φ) > 0 whenever φ ∈ (φ∗
g, 1). It then follows that the following statements are valid:
(1) If φ ∈ ³ λ1 r1+λ1, φ ∗ g ´
, then species 2 wins the competition. (2) If φ ∈ (φ∗
Case D: r1 < αK2 and r2 < βK1. In this case, if φ = 1, i.e., there is
no bad season, then the bistablity occurs for two species. Given φ ∈ (0, 1), we have g(φ) < 0; and f (φ) > 0 whenever φ ∈
³ λ1 r1+λ1, φ ∗ f ´ , and f (φ) < 0 whenever φ ∈ (φ∗
f, 1). It then follows that the following statements are valid:
(1) If φ ∈ ³ λ1 r1+λ1, φ ∗ f ´
, then species 2 wins the competition. (2) If φ ∈ (φ∗
f, 1), then the bistablity occurs for two species.
Acknowledgment. Xiao-Qiang Zhao would like to thank the National Cen-ter for Theoretical Science, National Tsing Hua University, Taiwan for its financial support and kind hospitality during his visit there.
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