The dynamics of a zooplankton–fish system in aquatic habitats

Nonlinear Analysis: Real World Applications 53 (2020) 103075

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

The dynamics of a zooplankton–fish system in aquatic habitats Yu Jin a ,∗, Feng-Bin Wang b,c a

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan c Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan b

article

info

Article history: Received 11 March 2019 Received in revised form 2 December 2019 Accepted 6 December 2019 Available online xxxx Keywords: Diel vertical migration zooplankton–fish system Predation avoidance Persistence Periodic semiflow

abstract Diel vertical migration is a common movement pattern of zooplankton in marine and freshwater habitats. In this paper, we use a temporally periodic reaction– diffusion–advection system to describe the dynamics of zooplankton and fish in aquatic habitats. Zooplankton live in both the surface water and the deep water, while fish only live in the surface water. Zooplankton undertake diel vertical migration to avoid predation by fish during the day and to consume sufficient food in the surface water during the night. We establish the persistence theory for both species as well as the existence of a time-periodic positive solution to investigate how zooplankton manage to maintain a balance with their predators via vertical migration. Numerical simulations discover the effects of migration strategy, advection rates, domain boundary conditions, as well as spatially varying growth rates, on persistence of the system. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Diel vertical migration (DVM) is a common movement pattern of zooplankton, such as copepods and Daphnia, in marine and freshwater habitats; see e.g., [1–9]. In this migratory behavior, individuals migrate downward from the warm surface waters to the cold deep waters at dawn and stay there during the daytime; they migrate upward from the deep waters to the surface waters at dusk and stay in the surface layer during the nighttime. Biologists have studied DVM for more than a century and there have been considerable discussions of the reason for DVM. The factors that might influence DVM of zooplankton include temperature, light intensity, ultraviolet radiation damage, food availability, nutritional state, migrants’ life stage, predator pressure, etc; see e.g., [8,10]. There were also arguments about whether DVM is demographic or metabolic advantage or disadvantage (see e.g., [6,7,11]). While discussions have not lead to a simple reason for DVM, more and more experimental evidences and field observations have ∗ Corresponding author. E-mail addresses: [email protected] (Y. Jin), [email protected] (F.-B. Wang).

https://doi.org/10.1016/j.nonrwa.2019.103075 1468-1218/© 2019 Elsevier Ltd. All rights reserved.

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Y. Jin and F.-B. Wang / Nonlinear Analysis: Real World Applications 53 (2020) 103075

shown that the DVM behavior of zooplankton seems to be an important strategy to avoid predation by visually orienting predators [2,4,5,10,11]. Zooplankters move to the deep water when their predators can easily find them in the surface water during the daytime and move back to the surface water during the nighttime to consume sufficient food (e.g., phytoplankton) and/or perform other physiological activities. The most benefit results from the trade-off between higher predation risk in surface waters and reduced growth in deeper waters [3]. As a result, DVM has been undoubtedly an important behavior that changes the dynamics of zooplankton, including their vertical and lateral distribution, and related ecosystems [1]. Mathematical models have been used to study DVM or predator prey models with predation avoidance movement. In [9], a habitat-selection game model between predator and prey in two water layers was proposed to explain the vertical migration. In [12], a one-dimensional advection equation was used to describe a possible control mechanism of DVM. The migration velocity was studied from the explicit relation with the light intensity, predation pressure, food availability, and temperature distribution. In [13], the authors discussed optimal trade-offs between benefits and costs involved in zooplanktonic DVM in different seasonal environments. In [14], a parameterization of DVM in the ocean was developed and integrated with a sizestructured food-web model to estimate the impact of DVM on marine ecosystems. In [15], the dynamics of temporally constant and periodic ordinary differential equations for a food chain (say, phytoplankton, zooplankton, and fish), incorporating migratory behavior of the middle predator, was studied. In [16], the authors studied a general two-patch predator–prey model where the transfer rates of the prey were assumed to depend on the densities of both prey and predator. To our knowledge, there has not been theoretical studies about DVM in spatial models. In this work, we will develop a reaction–diffusion–advection model to describe the dynamics of zooplankton and fish in aquatic habitats such as lakes or rivers. We divide the water column into two layers vertically: the surface water layer and the deep water layer. Zooplankton performs diel vertical migration, which is represented by the transformation of zooplankton population between the two layers. The upward migration (from deep water to surface water) rate depends on the light intensity while the downward migration (from surface water to deep water) rate depends on the light intensity and fish density. We will investigate how diel vertical (predation-avoidant) migration, together with other factors, influences the long-term dynamics of the zooplankton–fish system via theoretical and numerical studies of the model. Our model and results can also be applied to other systems with similar behaviors such as a fish-bird system, where fish may migrate to deep water to avoid predation by birds. The organization is as follows. In Section 2, we will establish the zooplankton–fish model. In Section 3, we will present the analysis of dynamics of the zooplankton system and the fish system, respectively. In Section 4, we will study the global stability of zooplankton-only and fish-only temporally periodic solutions and derive persistence theory for the whole system. We also establish the existence of a temporally periodic coexistence solution. In Section 5, we will investigate the influences of factors such as the diel vertical migration, the flow velocity, the domain boundary conditions, and the demographic parameters of zooplankton and fish on persistence via numerical simulations. A short discussion then completes the paper. 2. The model Assume that the aquatic habitat consists of the surface water zone and the bottom water zone. Both zones are represented by a one-dimensional interval [0, L], where L is the longitudinal length of the habitat. The populations in each zone are well mixed; the flow in each zone is at a steady state; the physical, hydrological, and biological conditions are the same throughout a cross-section x = x0 ∈ [0, L] in a given zone, where x is the spatial variable along the longitudinal direction of the habitat. By applying the conservation law, we can obtain the following model describing the dynamics of zooplankton (in two different zones) and fish in

Y. Jin and F.-B. Wang / Nonlinear Analysis: Real World Applications 53 (2020) 103075

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Fig. 2.1. A simple sketch of model (2.1).

the habitat (see Fig. 2.1): ⎧ ∂Z 1 ⎪ = ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂Z ⎨ 2 = ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂F ⎩ = ∂t

for x ∈ (0, L) and t > 0, L1 [Z1 ] + f1 (t, x, Z1 )Z1 − µ(t, x)M2 (Z1 ) + L2 [Z2 ] + f2 (t, x, Z2 )Z2 +

A2 (x) A1 (x) σ(t, x)M1 (Z2 , F ),

A1 (x) A2 (x) µ(t, x)M2 (Z1 )

− σ(t, x)M1 (Z2 , F )

(2.1)

− H(t, x, Z2 , F ), Lf [F ] + c(t)H(t, x, Z2 , F ) + θ(t, x)F − m(t, x)F − η(t, x)F 2 ,

with the spatial operators being defined as ∂ L1 := −v1 (x) ∂x +

1 ∂ ∂ A1 (x) ∂x D1 (x)A1 (x) ∂x , [ ] ∂ ∂ ∂ L2 := −v2 (x) ∂x D2 (x)A2 (x) ∂x , + A21(x) ∂x [ ] ∂ 1 ∂ ∂ Lf := −vf (x) ∂x + A2 (x) ∂x Df (x)A2 (x) ∂x ,

[

]

where Z1 (t, x) is the density of zooplankton at location x in the bottom water zone at time t, Z2 (t, x) is the density of zooplankton at location x in the surface water zone at time t, F (t, x) is the density of fish at location x in the surface water zone at time t, f1 and f2 are the per capita growth rates of zooplankton in the bottom and surface water zones, respectively, µ is the migration rate at which zooplankton in the bottom water enter the surface water, σ is the migration rate at which zooplankton return to the bottom water from the surface water, θ is the reproduction rate of fish due to other resources, m is the natural death rate of fish, η is the density-dependent death rate of fish, H is the functional response for predation of fish on zooplankton, c is the conversion rate, A1 and A2 are cross-sectional areas in the bottom water zone and in the surface water zone, respectively, D1 , D2 and Df are diffusion rates of zooplankton in different zones and fish, respectively, v1 , v2 , and vf are advection rates of zooplankton in different zones and fish, respectively. In particular, M1 represents the predation-avoidance migration of zooplankton from the surface water to the bottom water; M2 represents the density-dependent migration of zooplankton from the bottom water to the surface water. If the density of other food resources of fish is u(x) and the predation of them by fish follows the Holling type II response, then we can write θ(t, x) = θ1 (t, x)u(x)/(θ2 (t, x) + u(x)) for some functions θ1 and θ2 . Furthermore, the boundary value conditions for (2.1) are given as follows: ⎧ ⎪ ⎨αN N (t, 0) − β N ∂N (t, 0) = 0, t > 0, N = Z , Z , F, 1 2 1 1 ∂x (2.2) ⎪ ⎩αN N (t, L) + β N ∂N (t, L) = 0, t > 0, N = Z , Z , F, 1 2 2 2 ∂x where αiN ’s and βiN ’s are nonnegative constants with (αiN )2 + (βiN )2 ̸= 0. The initial value conditions are N(0, x) = N0 (x) ≥ 0, 0 ≤ x ≤ L, N = (Z1 , Z2 , F ), N0 = (Z10 , Z20 , F 0 ), where Z10 , Z20 and F 0 are initial distributions of zooplankton and fish populations, respectively.

(2.3)

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Throughout this paper, we assume that f1 , f2 , µ > 0, σ > 0, c > 0, θ ≥ 0, m > 0, η > 0, and H are continuous functions, which are periodic in t with a period ω > 0, and that Df , Di ’s, and Ai ’s are functions in C 2 ([0, L], (0, ∞)). Furthermore, we make the following assumptions for the growth, migration, and predation functions. C(i) For i = 1, 2, fi : [0, ∞) × [0, L] × [0, ∞) → R is monotonically decreasing and Liptschiz continuous with respect to Zi ; fi (t, x, 0) < ∞; there exist K1 > 0 and K2 > 0 such that for all x ∈ [0, L] and t ≥ 0, fi (t, x, Zi ) ≤ 0 for Zi ≥ Ki . Here K1 and K2 can be considered as the carrying capacity of ˜ 1 ≥ K1 and K ˜ 2 ≥ K2 zooplankton. Furthermore, we assume that there exist positive constants K such that ⎧ } { { } A2 (x) ⎪ ˜ 2 ≤ 0, ˜ ˜ ⎪ σ(t, x) K max f (t, x, K ) K + max ⎪ 1 1 1 ⎪ t∈[0,ω] A1 (x) ⎨ t∈[0,ω] x∈[0,L] x∈[0,L] { } { } A1 (x) ⎪ ˜ ˜ ˜ 1 ≤ 0. ⎪ max f (t, x, K2 ) K2 + max µ(t, x) K ⎪ ⎪ t∈[0,ω] ⎩ t∈[0,ω] 2 A2 (x) x∈[0,L]

x∈[0,L]

˜ 1 and K ˜ 2 will be used to guarantee that the solutions of (2.1)–(2.3) are ultimately bounded. For K ˜ 1 and K ˜ 2 follows from the existence of K1 and the functions we use in Section 5, the existence of K K2 . C(ii) M1 and M2 are nonnegative and differentiable functions. M1 (Z2 , 0) = 0, M1 (0, F ) = 0, M1 increases in Z2 and F , respectively, and M1 (Z2 , F ) ≤ Z2 for all Z2 ≥ 0 and F ≥ 0. M2 (0) = 0, M2 increases in Z1 , and M2 (Z1 ) ≤ Z1 . These assumptions indicate that zooplankton transfer more from the surface water to the bottom water if the zooplankton or fish densities increase in the surface water, that more zooplankton transfer from the bottom water to the surface water when the zooplankton density in the bottom water increases, and that the amount of transferable zooplankton is limited by the total amount of zooplankton in each zone, but the downward transfer does not occur if there is no fish. C(iii) H(t, x, Z2 , F ) is differentiable in x, Z2 and F , H(t, x, Z2 , 0) = 0, H(t, x, 0, F ) = 0, HZ2 ≥ 0, HF ≥ 0, HZ2 (x, Z2 , 0) = 0, and HF (x, 0, F ) = 0 for all x ∈ [0, L] and nonnegative Z2 and F . There exists KF > 0 such that c(t)H(t, x, Z2 , F ) + θ(t, x)F − m(t, x)F − η(t, x)F 2 < 0 for all t ≥ 0, x ∈ [0, L], Z2 ≥ 0, and F > KF . Note that Holling II, Holling III, and Beddington–DeAngelis functional responses satisfy these assumptions. n ˜ n = C([0, L], Rn ) and X ˜ n+ = C([0, L], R+ For n = 1, 2, 3, let X ) with norm ∥u∥ = max1≤i≤n maxx∈[0,L] ˜ ˜n = |ui (x)| for u = (u1 , . . . , un ) ∈ Xn in the case of Robin boundary conditions for (2.1)–(2.3) or X n ˜ n+ = C 1 ([0, L], R+ ) with norm ∥u∥ = max1≤i≤n (maxx∈[0,L] |ui (x)| + maxx∈[0,L] |u′i (x)|) C01 ([0, L], Rn ) and X 0 ˜ ˜ n+ is the for u = (u1 , . . . , un ) ∈ Xn in the case of Dirichlet boundary conditions for (2.1)–(2.3). Then X ˜ n with the above corresponding norm. positive cone in the Banach space X

3. Dynamics of the zooplankton system and the fish system In this section, we study the dynamics of the zooplankton system (i.e., when F = 0) and the fish system (i.e., when Z1 = Z2 = 0). 3.1. Dynamics of the zooplankton system When there is no fish in the surface water, zooplankton still transfer from the bottom water to the surface water but do not transfer back, as there is no predation pressure in the surface water where the habitat

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conditions are better. Letting F = 0 in (2.1)–(2.3), we have the following zooplankton system ⎧ ⎪ ∂Z1 ⎪ ⎪ ⎪ = L1 [Z1 ] + f1 (t, x, Z1 )Z1 − µ(t, x)M2 (Z1 ), x ∈ (0, L), t > 0, (a) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂Z2 = L2 [Z2 ] + f2 (t, x, Z2 )Z2 + A1 (x) µ(t, x)M2 (Z1 ), x ∈ (0, L), t > 0, (b) ⎪ ⎪ A2 (x) ⎪ ⎨ ∂t ∂Z ∂Z1 1 α1Z1 Z1 (t, 0) − β1Z1 (t, 0) = 0, α2Z1 Z1 (t, L) + β2Z1 (t, L) = 0, t > 0, (c) ⎪ ⎪ ⎪ ∂x ∂x ⎪ ⎪ ⎪ ⎪ ∂Z2 ∂Z2 ⎪ ⎪ α1Z2 Z2 (t, 0) − β1Z2 (t, 0) = 0, α2Z2 Z2 (t, L) + β2Z2 (t, L) = 0, t > 0, (d) ⎪ ⎪ ∂x ∂x ⎪ ⎪ ⎪ ⎪ ⎩ Zi (0, x) = Zi0 (x), 0 ≤ x ≤ L, i = 1, 2. (e)

5

(3.1)

The biologically relevant domain for system (3.1) is given by ˜ + : Z10 (x) ≤ K ˜ 1 , Z20 (x) ≤ K ˜ 2 on [0, L]}, X = {(Z10 , Z20 ) ∈ X 2

(3.2)

˜ 1 and K ˜ 2 are given in C(i). where K We first show that solutions of system (3.1) exist globally on [0, ∞) and converge to a compact attractor in X in the following Lemma with proof given in Appendix A. Lemma 3.1. For every initial value function N0 = (Z10 , Z20 ) ∈ X, system (3.1) has a unique solution (Z1 (t, ·), Z2 (t, ·)) on [0, ∞) with (Z1 (0, ·), Z2 (0, ·)) = N0 and (Z1 (t, ·), Z2 (t, ·)) ∈ X for all t ≥ 0. Furthermore, the periodic semiflow Φt : X → X generated by (3.1) is defined by Φt (N0 )(·) = (Z1 (t, ·), Z2 (t, ·)), t ⩾ 0, and Φt : X → X has a global compact attractor in X. Note that (3.1)(a) is decoupled. Linearizing (3.1)(a) with boundary conditions (3.1)(c) at Z1 = 0 yields ⎧ ⎨ ∂Z1 = L1 [Z1 ] + f1 (t, x, 0)Z1 − µ(t, x)M2,Z1 (0)Z1 , x ∈ (0, L), t > 0, (3.3) ∂t ⎩Z satisfies boundary conditions in (3.1)(c). 1 Substituting Z1 (t, x) = eλt φ1 (t, x) into (3.3), we obtain the eigenvalue problem ⎧ ⎪ ⎨φ1,t (t, x) = L1 [φ1 ] + (f1 (t, x, 0) − µ(t, x)M2,Z1 (0))φ1 (t, x) − λφ1 (t, x), x ∈ (0, L), φ1 satisfies boundary conditions in (3.1)(c), ⎪ ⎩ φ1 is ω -periodic in t.

(3.4)

By the Krein–Rutman Theorem (see, e.g., [17, Theorem7.2] and the arguments similar to those in [17, II.14], it follows that the eigenvalue problem (3.4) admits a principal eigenvalue, denoted by λ0 , which is associated ˜1. with a strongly positive ω-periodic eigenfunction in X By applying [18, Theorem 2.3.4] to the Poincar´e map associated with (3.1)(a), (3.1)(c), we obtain the following lemma. See also [19, Proposition 3.1]. Lemma 3.2. The following statements are true for any solution Z1 (t, x) of (3.1)(a), (3.1)(c) with initial ˜ + satisfying Z 0 (·) ≤ K ˜ 1. function Z10 in X 1 1 (i) If λ0 ≤ 0, then limt→∞ Z1 (t, x) = 0, uniformly for x ∈ [0, L]. ˜ + with Z ∗ (t, x) ≤ (ii) If λ0 > 0, then (3.1)(a), (3.1)(c) admits a unique positive ω-periodic solution Z1∗ ∈ X 1 1 ∗ ˜ 1 such that limt→∞ (Z1 (t, x) − Z (t, x)) = 0 uniformly for x ∈ [0, L] provided that Z 0 ̸≡ 0. K 1

1

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When Z1 = 0, (3.1)(b), (3.1)(d) becomes ⎧ ⎨ ∂Z2 = L [Z ] + f (t, x, Z )Z , x ∈ (0, L), t > 0, 2 2 2 2 2 ∂t ⎩ Z2 satisfies boundary conditions in (3.1)(d).

(3.5)

The eigenvalue problem associated with the linearization of (3.5) at Z2 = 0 is ⎧ ⎪ ⎨φ2,t (t, x) = L2 [φ2 ](x) + f2 (t, x, 0)φ2 (t, x) − λφ2 (t, x), x ∈ (0, L), φ2 satisfies boundary conditions in (3.1)(d), ⎪ ⎩ φ2 is ω -periodic in t.

(3.6)

¯ associated with a strongly positive ω-periodic Similarly as (3.4), (3.6) admits a unique principal eigenvalue λ ˜ 1 . Moreover, we obtain the following results by similar mathematical arguments as for eigenfunction in X Lemma 3.2. Lemma 3.3.

˜ + satisfying Z 0 (·) ≤ K ˜ 2. Let Z2 (t, x) be the solution of (3.5) with initial condition Z20 in X 2 1

¯ ≤ 0, then limt→∞ Z2 (t, x) = 0 uniformly for x ∈ [0, L]. (i) If λ ¯ > 0, then there exists a unique positive ω-periodic solution Z¯2 (t, x) of (3.5) (with Z¯2 ∈ X ˜ + and (ii) If λ 1 ˜ 2 ) such that limt→∞ (Z2 (t, x) − Z¯2 (t, x)) = 0 uniformly for x ∈ [0, L] provided that Z 0 ̸≡ 0. Z¯2 (t, x) ≤ K 2

When Z1 = Z1∗ , (3.1)(b), (3.1)(d) becomes ⎧ ⎨ ∂Z2 = L2 [Z2 ] + f2 (t, x, Z2 )Z2 + A1 (x) µ(t, x)M2 (Z ∗ (t, x)), 1 ∂t A2 (x) ⎩ Z2 satisfies boundary conditions in (3.1)(d).

(3.7)

Similarly as in [20, Lemma 3.1], we can prove that the Poincar´e map Qω associated with (3.7) is strongly positive, strongly monotone, strictly subhomogeneous, point dissipative, uniformly bounded, and compact ˜ 2] ⊂ X ˜ 2 are constant functions in X ˜ + where 0 and K ˜ + with values 0 and K ˜ 2 , respectively. on [0, K 1

1

Moreover, by C(i) we can obtain that Qω is weakly uniformly persistent. Then by similar arguments as in [20, Lemma 3.1]. we can prove that Qω admits a unique positive fixed point that is globally asymptotically ˜ 2 ]\{0}, which indicates that (3.7) admits a unique positive ω-periodic solution Z ∗ (t, x), which stable in [0, K 2 ˜ 2 ] \ {0}. is globally asymptotically stable for all initial conditions in [0, K Therefore, we have the following results for the dynamics of model (3.1) with proof in Appendix B. Lemma 3.4. Let (Z1 (t, x), Z2 (t, x)) = (Z1 (t, x, N0 ), Z2 (t, x, N0 )) be the solution of (3.1) with initial function N0 = (Z10 , Z20 ) ∈ X. ¯ ≤ 0, then limt→∞ (Z1 (t, x), Z2 (t, x)) = (0, 0), uniformly for x ∈ [0, L]. (i) If λ0 ≤ 0 and λ ¯ > 0, then there exists a unique positive ω-periodic function Z¯2 (t, x) such that (0, Z¯2 (t, x)) (ii) If λ0 ≤ 0 and λ is a solution of (3.1), and lim ((Z1 (t, x), Z2 (t, x)) − (0, Z¯2 (t, x))) = 0,

t→∞

uniformly for x ∈ [0, L] provided that Z20 ̸≡ 0. (iii) If λ0 > 0, then (3.1) admits a unique positive ω-periodic solution (Z1∗ (t, x), Z2∗ (t, x)) such that limt→∞ ((Z1 (t, x), Z2 (t, x)) − (Z1∗ (t, x), Z2∗ (t, x))) = 0, uniformly for x ∈ [0, L] provided that Z10 ̸≡ 0.

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3.2. Dynamics of the fish system When there is no zooplankton in the habitat, fish live on other food resources. Letting Z1 = Z2 = 0 in (2.1), we obtain the following system: ⎧ ∂F ⎪ ⎪ = Lf [F ] + θ(t, x)F − m(t, x)F − η(t, x)F 2 , x ∈ (0, L), t > 0, (a) ⎪ ⎨ ∂t ∂F ∂F (3.8) (t, 0) = 0, α2F F (t, L) + β2F (t, L) = 0, t > 0, (b) α1F F (t, 0) − β1F ⎪ ⎪ ∂x ∂x ⎪ ⎩ F (0, x) = F 0 (x), 0 ≤ x ≤ L. (c) Let F¯ :=

( max t∈[0,ω],x∈[0,L]

θ(t, x) − m(t, x) η(t, x)

) .

˜ in X ˜ > F¯ . Thus, the ˜ + is an upper solution of (3.8) when M It is not hard to see that a constant function M 1 solutions of system (3.8) are eventually bounded if the initial conditions are nonnegative. Next, we consider the following eigenvalue problem ⎧ ⎪ ⎨ϕt (t, x) = Lf [ϕ] + [θ(t, x) − m(t, x)]ϕ(t, x) − ζϕ(t, x), x ∈ (0, L), (3.9) ϕ satisfies boundary conditions in (3.8)(b), ⎪ ⎩ ϕ is ω -periodic in t. It is easy to see that, similar as (3.4), (3.9) admits a principal eigenvalue ζ 0 associated with a strongly ˜ 1 . Moreover, we obtain the following results whose mathematical arguments are positive eigenfunction in X similar to Lemma 3.2. ˜ + and F 0 (·) ≤ F¯ . Lemma 3.5. Suppose that F (t, x) is the unique solution of system (3.8) with F 0 ∈ X 1 Then the following results are valid. (i) If ζ 0 ≤ 0, then limt→∞ F (t, x) = 0, uniformly for x ∈ [0, L]. (ii) If ζ 0 > 0, then there exists a unique positive ω-periodic solution F˜ (t, x) such that limt→∞ (F (t, x) − F˜ (t, x)) = 0, uniformly for x ∈ [0, L], provided that F 0 ̸≡ 0. 4. Coexistence of the zooplankton–fish system In this section, we will investigate the dynamics of the full system (2.1). In particular, we will establish conditions for either zooplankton or fish to persist or for both species to coexist in the habitat. The biologically relevant domain for system (2.1) is given by ˜ + : Z10 (x) ≤ K ˜ 1 , Z20 (x) ≤ K ˜ 2 for x ∈ [0, L]}, Y = {(Z10 , Z20 , F 0 ) ∈ X 3

(4.1)

˜ 1 and K ˜ 2 are given in C(i). where K We first show that solutions of system (2.1) exist globally on [0, ∞) and converge to a compact attractor in Y. The proof is in Appendix C. Lemma 4.1. For every initial value function v0 ∈ Y, system (2.1) has a unique solution (Z1 (t, ·), Z2 (t, ·), F (t, ·)) on [0, ∞) with (Z1 (0, ·), Z2 (0, ·), F (0, ·)) = v0 and (Z1 (t, ·), Z2 (t, ·), F (t, ·)) ∈ Y for all t ≥ 0. The periodic semiflow Ψt : Y → Y generated by (2.1) is defined by Ψt (v0 )(x) = (Z1 (t, x), Z2 (t, x), F (t, x)), t ⩾ 0, and Ψt : Y → Y has a global compact attractor in Y.

(4.2)

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The following result shows the strong positivity of solutions of system (2.1). The proof is in Appendix D. Lemma 4.2. Let (Z1 (t, x), Z2 (t, x), F (t, x)) be the solution of system (2.1) with initial value functions (Z1 (0, ·), Z2 (0, ·), F (0, ·)) ∈ Y. (i) If there is a t0 ⩾ 0 such that Z1 (t0 , ·) ̸≡ 0, then Z1 (t, x) > 0 for t > t0 and all x in [0, L] other than the Dirichlet boundary points. (ii) If there is a t0 ⩾ 0 such that Z2 (t0 , ·) ̸≡ 0, then Z2 (t, x) > 0 for t > t0 and all x in [0, L] other than the Dirichlet boundary points. (iii) If there is a t0 ⩾ 0 such that F (t0 , ·) ̸≡ 0, then F (t, x) > 0 for t > t0 and all x in [0, L] other than the Dirichlet boundary points. By Lemmas 3.4 and 3.5, we conclude that system (2.1) has the following trivial and semi-trivial ω-periodic solutions: (i) (ii) (iii) (iv)

Trivial solution E0 = (0, 0, 0), always existing; ¯ > 0; ¯ x) = (0, Z¯2 (t, x), 0), existing if λ Semi-trivial ω-periodic solution E(t, ∗ ∗ ∗ Semi-trivial ω-periodic solution E (t, x) = (Z1 (t, x), Z2 (t, x), 0), existing if λ0 > 0; ˜ x) = (0, 0, F˜ (t, x)), existing if ζ 0 > 0. Semi-trivial ω-periodic solution E(t,

In the rest of this section, we will study the stability of the above solutions and uniform persistence of (2.1). We first introduce a few more principal eigenvalues but omit the proof of their existence as it is similar to that for (3.4). Let ζ¯ be the principal eigenvalue of the eigenvalue problem associated with the linearized ¯ x): system of (2.1) at E(t, ⎧ ¯ ⎪ ⎨ϕt (t, x) = Lf [ϕ] + [c(t)HF (t, x, Z2 , 0) + θ(t, x) − m(t, x)]ϕ(t, x) − ζϕ(t, x), x ∈ (0, L), (4.3) ϕ satisfies boundary conditions in (3.8)(b), ⎪ ⎩ ϕ is ω -periodic in t. Let ζ ∗ be the principal eigenvalue of the eigenvalue problem associated with the linearized system of (2.1) at E ∗ (t, x): ⎧ ∗ ⎪ ⎨ϕt (t, x) = Lf [ϕ] + [c(t)HF (t, x, Z2 , 0) + θ(t, x) − m(t, x)]ϕ(t, x) − ζϕ(t, x), x ∈ (0, L), (4.4) ϕ satisfies boundary conditions in (3.8)(b), ⎪ ⎩ ϕ is ω -periodic in t. Let Λ0 be the principal eigenvalue of the eigenvalue problem associated with the linearized system of (2.1) ˜ x): at E(t, ⎧ A2 (x) ⎪ ⎪ φ1,t (t, x) = L1 [φ1 ] + f1 (t, x, 0)φ1 − µ(t, x)M2,Z1 (0)φ1 + σ(t, x)M1,Z2 (0, F˜ )φ2 ⎪ ⎪ A ⎪ 1 (x) ⎪ ⎪ ⎪ −Λφ1 (t, x), x ∈ (0, L), t > 0, ⎪ ⎪ ⎪ ⎨ A1 (x) φ2,t (t, x) = L2 [φ2 ] + f2 (t, x, 0)φ2 + µ(t, x)M2,Z1 (0)φ1 − σ(t, x)M1,Z2 (0, F˜ )φ2 (4.5) A2 (x) ⎪ ⎪ ⎪ ˜ ⎪ −HZ2 (t, x, 0, F )φ2 − Λφ2 (t, x), x ∈ (0, L), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ (φ , φ ) satisfies boundary conditions in (3.1)(c)–(3.1)(d), 1 2 ⎪ ⎪ ⎩ (φ1 , φ2 ) is ω -periodic in t. Remark 4.1. By the theory of eigenvalue problems for differential operators (see e.g., [17, Lemma 15.5]), we know that ζ¯ > ζ 0 and ζ ∗ > ζ 0 if ζ¯ and/or ζ ∗ exists. By adapting the proof of [17, Lemma 15.5] to eigenvalue problems (3.4) and (4.5) restricted to φ1 , we can also obtain that λ0 < Λ0 if Λ0 exists.

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We then can obtain the following results about the stability of all trivial and semi-trivial ω-periodic solutions. Proposition 4.1. ¯ ≤ 0, and ζ 0 ≤ 0, then E0 is locally asymptotically stable; otherwise, E0 is unstable. (i) If λ0 ≤ 0, λ ¯ > 0 and ζ¯ < 0, then E(t, ¯ x) is locally asymptotically stable; (ii) If λ0 ≤ 0, and λ ¯ otherwise, E(t, x) is unstable. (iii) If λ0 > 0 and ζ ∗ < 0, then E ∗ (t, x) is locally asymptotically stable; otherwise, E ∗ (t, x) is unstable. ˜ x) is locally asymptotically stable; otherwise, E(t, ˜ x) is unstable. (iv) If ζ 0 > 0 and Λ0 < 0, then E(t, Let

W0 = {v0 = (Z10 , Z20 , F 0 ) ∈ Y : Z10 ̸≡ 0, Z20 ̸≡ 0, and F 0 ≡ ̸ 0}, ∂W0 = Y\W0 = {v0 = (Z10 , Z20 , F 0 ) ∈ Y : Z10 ≡ 0, or Z20 ≡ 0, or F 0 ≡ 0}.

In the following, we will prove a result about the uniform persistence and coexistence of zooplankton and fish for system (2.1). The proof is in Appendix E. Theorem 4.1. Let (Z1 (t, x, v0 ), Z2 (t, x, v0 ), F (t, x, v0 )) be a solution of (2.1) with initial condition (Z1 (0, ·, v0 ), Z2 (0, ·, v0 ), F (0, ·, v0 )) = v0 = (Z10 , Z20 , F 0 ) ∈ Y. Assume one set of the following conditions is true: (i) (ii) (iii) (iv)

λ0 λ0 λ0 λ0

> 0, ζ 0 > 0, and Λ0 > 0, > 0 and ζ 0 < 0 < ζ ∗ , ¯ > 0, ζ 0 > 0, and Λ0 > 0, < 0, λ ¯ > 0 and ζ 0 < 0 < ζ. ¯ < 0, λ

Then there exists ξ0 > 0 such that for any v0 ∈ W0 , we have lim inf Zi (t, x, v0 ) ≥ ξ0 , and lim inf F (t, x, v0 ) ≥ ξ0 , ∀ i = 1, 2, x ∈ [0, L], t→∞

t→∞

(4.6)

in the case of Robin boundary conditions or lim inf Zi (t, x, v0 ) ≥ ξ0 e˜(x), and lim inf F (t, x, v0 ) ≥ ξ0 e˜(x), ∀ i = 1, 2, x ∈ [0, L], t→∞

t→∞

(4.7)

˜ + ). Moreover, in the case of Dirichlet boundary conditions, where e˜(·) is a given positive element in Int(X 3 p p p system (2.1) admits at least one positive ω-periodic solution (Z1 (t, x), Z2 (t, x), F (t, x)). 5. Influences of parameters on population persistence Proposition 4.1 provides conditions for none species or only one species to persist in the habitat. Theorem 4.1 provides conditions for uniform persistence and coexistence of both zooplankton and fish. In the worst situation, if zooplankton cannot survive in the whole habitat and if fish cannot survive on other food resources, then both species will eventually be extinct. Regarding persistence of one species, we have the following observations from Proposition 4.1 and Theorem 4.1: (ZP) zooplankton persist in the whole habitat if it can persist in the bottom water when subjected to upward migration (i.e., λ0 > 0) or if it can grow sufficiently in the surface water without being eradicated by ¯ > 0, and ζ 0 < 0 or ζ 0 > 0 and Λ0 > 0); fish (i.e., λ0 < 0, λ (FP) fish persists in the habitat if there are enough other food resources (i.e., ζ 0 > 0) or if there are enough zooplankters in the surface water or in the bottom water (i.e., ζ 0 < 0 < ζ ∗ and λ0 > 0, and/or ¯ > 0). ζ 0 < 0 < ζ¯ and λ

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Uniform persistence and coexistence of both zooplankton and fish can only occur when both of the following conditions are valid (see Theorem 4.1): (UP1) zooplankton can persist in the bottom water allowing upward migration (i.e., λ0 > 0) or in the surface ¯ > 0); water without fish or migration (i.e., λ ¯ ∗) (UP2) fish survive on zooplankton or other resources but do not eradicate zooplankton (i.e., ζ 0 < 0 < ζ(ζ or ζ 0 > 0 and Λ0 > 0). In this section, we conduct numerical simulations to study the influences of different factors on persistence of zooplankton, fish or both species in the habitat. For simplicity, we first assume that all parameters are spatially homogenous. We choose logistic type growth functions for zooplankton: ) ( ) ( Z2 Z1 , f2 (t, Z2 ) = r2 (t) 1 − , f1 (t, Z1 ) = r1 (t) 1 − K1 (t) K2 (t) where ri (i = 1, 2) is the intrinsic growth rate and Ki (i = 1, 2) is the carrying capacity for zooplankton. Observations have found that vertical migration becomes significant when the fish density is above a certain level and stops increasing when the fish density is above a higher level (see e.g., [9]). We choose the migration functions as F2 M1 (Z2 , F ) = 2 Z2 , M2 (Z1 ) = Z1 , m1 + F 2 where the predation avoidance (downward) migration M1 is small when the fish density is small and quickly becomes large when the fish density is sufficiently large, and the upward migration M2 only depends on the zooplankton density in the bottom. We use the light intensity at the surface water zone as suggested in [21]: ) ⏐ (π )⏐) 1( (π ⏐ ⏐ (t − 6) + ⏐sin (t − 6) ⏐ , I0 = sin 2 12 12 where t is expressed in hours and the daytime length is 12 h, while the maximum light intensity is assumed to occur at noon (12 : 00 h). The migration rates are chosen as µ(t) = µ0 (1 − I0 (t)) + µ1 , σ(t) = σ0 I0 (t) + σ1 .

(5.1)

It has been found that the upward and downward migration is a constant behavior that always exists in daytime and nighttime (see e.g., [11]), so we assume small positive lower bounds µ1 and σ1 for our choices of µ and σ. At noon, the upward migration rate µ attains its minimum and the downward migration rate σ attains its maximum, while at midnight, the upward migration rate µ attains its maximum and the downward migration rate σ attains its minimum. The predation follows Holling type II functional response H(t, Z2 , F ) =

b(t)Z2 F, 1 + ab(t)Z2

where the handling time a is a constant and the searching rate b(t) depends on the light intensity because we assume that fish only forage by sight. In the rest of this section, according to references such as [14,22,23] and our best understanding, the following parameter values (forms) are used except otherwise specified: ω = 24 (h), L = 500 (m), A1 /A2 = 1, D1 = D2 = 360 (m2 /h), Df = 1800 (m2 /h), v1 = v2 = vf = 1.8 (m/h), r1 = 0.0271 (1/h), r2 = 0.4167 (1/h), K1 = K2 = 1 (kg/m3 ), θ = 0.05 (1/h), m = 0.0042 (1/h), η = 0.0208 (1/(kg·h)), m1 = 0.1 (kg/m3 ), µ1 = 0.0001 (1/h), σ1 = 0.0001 (1/h), a = 2, b(t) = 0.9I0 (t), c = 0.1. Three types of boundary conditions are used in simulations: (H) hostile condition: N (x) = 0, (ZF) zero-flux condition: DNx (x) − vN (x) = 0, (FF) free-flow condition: Nx (x) = 0,

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¯ = 0.4073, ζ¯ = 0.0538, Fig. 5.1. The value of Λ0 in the µ0 -σ0 plane. Left: η = 0.0208; right: η = 0.00416. Persistence quantities: λ ζ 0 = 0.0416, λ0 > 0 when µ0 < 0.0315 and λ0 < 0 when µ0 > 0.0315, and ζ ∗ increases in µ0 with values around 0.0538.

where N = Z1 , Z2 , or F , D = D1 , D2 or Df , v = v1 , v2 or vf , and x = 0 or L. The zero-flux upstream boundary condition and the free-flow downstream boundary condition are applied in simulations except otherwise specified. Influence of diel vertical migration We are most interested in how diel vertical migration influences population dynamics. Note that persistence dynamics of zooplankton or fish are descried by different principal eigenvalues of corresponding eigenvalue problems (see Proposition 4.1 and Theorem 4.1 or conditions (ZP), (FP), and (UP1)-(UP2)). Definitions of these eigenvalues indicate that the upward migration (i.e., µ(t, x)M2 (Z1 )) affects λ0 , ζ ∗ and Λ0 , and that the downward migration (i,e, σ(t, x)M1 (Z2 , F )) only affects Λ0 . By Theorem 4.1, the upward migration has a significant influence on zooplankton’s persistence in the bottom water and hence uniform persistence of the whole system, while the predator-avoidant downward migration, together with the upward migration, influences the coexistence of the two species specifically in the case where zooplankton can grow well in the surface water where there are lots of fish. Therefore, our theoretical results coincide with biological observations and experiments: diel vertical migration is a useful and necessary strategy for prey populations to persist with their predators in the aquatic environments. By applying the theory of periodic eigenvalue problems in [17, Lemma 15.5] to model (3.4), we know that 0 λ decreases with the upward migration rate µ0 . In our simulations, ζ ∗ , if exists, does not seem to vary much in µ0 . The relation between Λ0 and migration rates is more interesting. When the zero-flux upstream boundary condition and the free-flow downstream boundary condition are applied, Λ0 always increases with σ0 ; it decreases in µ0 when σ0 is small; it decreases and then increases in µ0 when σ0 is medium; it increases or first increases and then decreases in µ0 when σ0 is large; and it is large at large µ0 and σ0 values although the maximum of Λ0 may not be attained at the largest µ0 or σ0 ; see Fig. 5.1. In the example shown in Fig. 5.1, ¯ > 0, ζ¯ > 0, and ζ ∗ > 0. Therefore, fish always persists but zooplankton persist if Λ0 > 0 (see ζ 0 > 0, λ Theorem 4.1(i,iii)). Thus, we can study the sign of Λ to determine the coexistence of zooplankton and fish. We see that overall large vertical migration rates lead to large Λ0 and hence help zooplankton persist with fish. In particular, the more downward migration the better it is for zooplankton to persist. If the downward migration is low but the upward migration high, then more zooplankters are subjected to predation in the surface water and it is easier for zooplankton to be extinct. In other simulations, we also find that under the circumstance where the growth rate of zooplankton in the surface water r2 becomes smaller, zooplankters should migrate more to the bottom water in order to avoid predation and hence to coexist with fish (i.e., σ0 needs to be even larger for Λ0 to be positive). The left figure in Fig. 5.1 shows that, when the density dependence of fish (i.e., η) is high, higher upward and downward migration makes better condition for coexistence. On the other hand, when the density

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Fig. 5.2. The relations between λ0 (in blue), ζ ∗ (in green) and Λ0 (in red) with the migration rates. Here, r2 = 0.3542. The solid curves correspond to µ = µ0 (1 − I0 (t)) and σ = σ0 I0 (t) (case (a)); the “∗” curves correspond to µ =average(µ0 (1 − I0 (t))) and ¯ = 0.346, ζ¯ = 0.0538, σ =average(σ0 I0 (t)) (case (b)); the “+” curves correspond to µ = µ0 and σ = σ0 (case (c)). Other values: λ and ζ 0 = 0.0416. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

dependence of fish becomes smaller, the density of fish in the surface water becomes larger, and hence, zooplankters are subjected to stronger predation pressure in the surface water. In this case, the overall chance for zooplankton to coexist with fish becomes smaller so the Λ0 values in the right figure of Fig. 5.1 are much smaller than those in the left figure of Fig. 5.1, and moreover, very small upward and downward migration can also help zooplankton persist with fish in the whole habitat, if the bottom water is not too bad for zooplankton to grow (λ0 > 0 for very small µ0 ). In both cases, small downward migration but large upward migration is the worst strategy for zooplankton to persist with fish. We also used M1 = Z2 F/(m1 + F ) in simulations and obtained similar results. To study the influence of temporal fluctuations of migrations on persistence of zooplankton and fish, we consider three different types of migrations: (a) temporally periodic upward and downward migrations (i.e. diel vertical migration as in (5.1)), (b) constant migration at average rates (of the periodic migrations in (a)), and (c) constant migration at the maximum rates (of the periodic migrations in (a)); see Fig. 5.2. It turns out that the value of ζ ∗ does not change much in three situations; λ0 is the same in cases (a) and (b) but becomes much smaller in case (c); when µ0 and σ0 are small, the smallest Λ0 occurs in case (a) and the largest Λ0 occurs in case (c), which reverses when µ0 and σ0 are large. Therefore, performing diel vertical migration or constant averaged migration results in similar long-term persistence dynamics of zooplankton in the bottom water and the more migration the worse it is for persistence there (see values of λ0 ). In terms of coexistence of zooplankton and fish, diel vertical migration is the best strategy when the overall migration rates are large (see values of Λ0 ). Influence of boundary conditions Previous work showed that a single species persists better in a river with zero-flux or free-flow boundary conditions than in a river with hostile conditions (e.g., [24]). This does not seem to be necessarily true for zooplankton to coexist with fish in a predator–prey system as the Λ0 value is overall smaller in Fig. 5.1 where the zero-flux upstream condition and the free-flow downstream condition are applied than in Fig. 5.3 where at least one of the boundary conditions is hostile. When the populations are subjected to lethal boundary conditions, there is a trade-off for zooplankton among predation risk in the surface water, low growth in the bottom water, and bad boundary conditions. Fig. 5.3 shows that low upward migration (µ0 ) and high

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Fig. 5.3. The value of Λ0 in the µ0 -σ0 plane. Boundary conditions: left: hostile condition at x = 0 and x = L; middle: zero-flux condition at x = 0 and hostile condition at x = L; right: hostile condition at x = 0 and free-flow condition at x = L.

Fig. 5.4. Regions in the v1 -v2 plane for persistence. Below the line or on the left of the line represents the region where the corresponding quantity is positive, except that Λ0 is positive in region IV but negative below region IV. Here µ0 = 0.01, σ0 = 0.04.

downward migration (σ0 ) seem to be a bad strategy that may cause extinction of zooplankton more easily. This is probably because more individuals lose the opportunity to grow faster in the surface water while they are subjected to predation and boundary loss. Figs. 5.1 and 5.3 also indicate that if the upstream boundary is harmless (i.e., zero-flux upstream condition), then low downward migration cannot be the best option for zooplankton to persist, and that a good management of diel vertical migration may help zooplankton coexist well with fish in a habitat with lethal boundaries. Influence of flow advection Flow advection can be an important constraint on zooplankton’s growth and spatial and temporal ¯ ζ 0 , ζ, ¯ and ζ ∗ all distribution when they perform diel vertical migration [1]. Simulations show that λ0 , λ, decrease with v1 or v2 (= vf ), which indicates that overall large advection rates make the habitat worse for one or both populations to persist and that when the advection is sufficiently high in the whole habitat, both populations will be washed out. We also see that Λ0 decreases in v1 and increases slightly in v2 when v2 is small, which indicates that a slightly large advection in the surface water may prevent the growth of fish and hence help coexistence of both species. We then divide the first quadrant of the v1 -v2 plane according to the signs of all related principal eigenvalues in Proposition 4.1 and Theorem 4.1; see Fig. 5.4. In particular, regions I, II, III, and IV correspond to the conditions (i), (ii), (iii), and (iv) in Theorem 4.1, respectively. In this example, when the advection rates are small in both the bottom water and the surface water, both species can persist alone or together with each other (region I); when the washout effect becomes larger in the surface water, fish cannot survive only with other food resources, but can persist if they also feed

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on zooplankton (region II); when the advection rate is large in the bottom water but small in the surface water, zooplankton cannot persist just in the bottom water but can persist in the surface water, and moreover coexist with fish if other food resource of fish is not too much (regions III and IV). When v1 = v2 = 0, the habitat is a lake and the free-flow boundary condition is the zero-flux one. Fig. 5.4 shows that a zooplankton–fish system may uniformly persist in a lake but cannot coexist in a river when subjected to the zero-flux upstream and free-flow downstream boundary conditions. A positive periodic solution of (2.1) in a spatially varying habitat In Theorem 4.1, we established the existence of a positive periodic solution when model (2.1) is uniformly persistent. It is difficult to prove the stability or attractivity of this periodic solution, but in an example shown in Fig. 5.5, we solve (2.1) numerically and show that it is possible that the system will be eventually stabilized at the periodic solution, in the case where fish can persist without zooplankton (see the top and middle panels of Fig. 5.5), or where fish cannot persist without zooplankton (see the bottom panel of Fig. 5.5). The effect of vertical migration can be easily seen from the periodic solution (in the middle or bottom panel of Fig. 5.5): in the middle of daytime, zooplankton density reaches the maximum in the bottom water but minimum in the surface water, while in the middle of nighttime, its density reaches maximum in the surface water but minimum in the bottom water. The fish density does not vary much during a day but is higher during the daytime than at night. In this example, λ0 = −0.0031 < 0 if µ = 0, which implies that without upward migration and fish, zooplankton cannot persist in the bottom water. However, when fish appears, diel vertical migration helps zooplankton persist not only in the surface water but also in the bottom water. By comparing the middle panel and the bottom panel of Fig. 5.5, we see that if the other food supply becomes lower (i.e., when θ is smaller), then the fish density decreases but the zooplankton densities increase. Here we also investigate the effect of the spatial heterogeneity on the solution of (2.1). The growth rates of zooplankton in both the bottom water and the surface water are assumed to be high at the upstream and low at the downstream, that is, r1 and r2 are decreasing functions in x. It turns out that in the periodic solutions, the density of zooplankton is also high at the upstream and low at the downstream, either in the bottom water or in the surface water, but it may not attain the maximum at the upstream end. However, the spatial distribution of the fish density does not follow the pattern of the spatial distribution of the zooplankton densities and it is high at the downstream in both cases. 6. Discussion Richness and complexity of ecological communities in aquatic environments under temporal fluctuations and spatial variations are important in ecology. It has been shown that the features such as light intensity, oxygen concentration, temperature, substrates, and nutrients, vary largely at different water depth levels in aquatic habitats; see e.g., [25–27]. As a result, different species may prefer to live at different water depth levels or transfer between levels [25,26]. We proposed a temporally periodic partial differential equation system that incorporates spatial heterogeneities to describe the dynamics of a predator–prey system for fish and zooplankton in aquatic habitats. In our model, fish only live in the surface water level and catch zooplankters by vision; zooplankton access more food in the surface water but they perform diel vertical migration in order to avoid predation by fish. These features lead to the assumptions that the upward and downward migrations and the functional response are time-periodic as they all depend on the light intensity. We established existence and stability of the trivial and semi-trivial time-periodic solutions, existence of a global attractor, uniform persistence, and existence of a positive time-periodic solution of the whole system. By using these results, we obtained the persistence conditions for one or both species: zooplankton can persist in the whole habitat if they can persist in the bottom water when subjected to upward migration or if they can grow well in the surface water when subjected to predation by fish; fish can persist if there is enough other food or if there are enough zooplankters; coexistence of zooplankton and fish only occurs

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Fig. 5.5. Top panel: a solution of (2.1) when θ = 0.05; middle panel: the positive periodic solution of (2.1) that the solution in the top panel converges to; here ζ 0 = 0.0416, ζ¯ = 0.0538, Λ0 = 0.1005. Bottom panel: the positive periodic solution of (2.1) when θ = 0.0083; here ζ 0 = −6.9438 × 10−5 , ζ¯ = 0.0122. Other parameters and values: r1 = 0.0021 − 0.0001x, r2 = 0.4167 − 0.0008x, ¯ = 0.336. µ0 = σ0 = 0.5, λ0 = −0.366, λ

when the growth rate of zooplankton in one of the water layers is high and fish have enough food but do not eat up all zooplankton. Our theoretical and numerical results confirm the biological observation that diel vertical migration is an important behavior that influences zooplankton’s daily distribution and long-term dynamics in aquatic habitats, when they attempt to find a trade-off between high predation risk in the surface water and low growth rate in the bottom water (see e.g., [8]). Under such migration, zooplankton concentrate in the surface water during the nighttime and in the bottom water during the daytime (see Fig. 5.5). If diel vertical migration occurs, large upward migration during the nighttime and large downward migration during the daytime seem to be a reasonable strategy that helps zooplankton persist well with fish. If one part of the habitat boundaries is lethal, then zooplankton should also consider the boundary loss and in this situation large downward migration may not always benefit the persistence of two species. Similarly as for previous

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river population models, we find that large advection in both the surface water and the bottom water always washes out both zooplankton and fish, while small advection may allow persistence of the two species. However, both species may also coexist when they are subjected to large advection in the bottom water but not too large advection in the surface water, provided that zooplankton can grow sufficiently well in the surface water. Moreover, we also find that the spatial heterogeneity of growth rates of zooplankton influence the spatial distribution of zooplankton in the periodic solutions, but the spatial distribution of fish may not follow the same pattern as in the distributions of zooplankton. This is probably because the influence of the difference in the spatial resource distribution is not as significant as that of the boundary conditions for fish in the situation in concern. Our model and results can be applied to other predator–prey systems where the prey transfers in two subhabitats due to predator-avoidance while the predator only occupies one. For example, fish and invertebrates can live in a large range of water depth levels, while their predator – birds – can only catch their preys in the surface water zone of a deep lake (see e.g., [28]). In our work we divided the habitat vertically into the surface water zone and the bottom water zone to study the effect of the diel vertical migration. In real aquatic habitats, the temperature, the food abundance, and the predation pressure of zooplankton vary continuously with light intensity at different water depth level (see e.g., [12,13]). It would be interesting to include such vertical variations into mathematical models to further study diel vertical migration in the future. Acknowledgment The authors greatly appreciated valuable discussions with Professor Sze-Bi Hsu. We are also grateful to anonymous referees for their careful reading and helpful suggestions which led to significant improvements of our original manuscript. Y. J. was supported by National Science Foundation, USA (NSF DMS 1411703). Research of F.-B. Wang was supported in part by Ministry of Science and Technology, Taiwan; and National Center for Theoretical Sciences, Taiwan, National Taiwan University; and Chang Gung Memorial Hospital, Taiwan (CRRPD3H0011, BMRPD18, NMRPD5J0201 and CLRPG2H0041). Y. J. also acknowledged the hospitality of National Center for Theoretical Sciences (NCTS) in Taiwan while she visited there in summers 2016 and 2018. Appendix A. Proof of Lemma 3.1 By [29, Theorem 1 and Remark 1.1] and the similar arguments in the proof of [30, Lemma 2.1], we can show that system (3.1) has a unique noncontinuable solution and the solutions to (3.1) remain nonnegative on their interval of existence if they are nonnegative initially. It follows from assumption C(i) that (Z1 , Z2 ) = ˜ 1, K ˜ 2 ) is an upper solution of (3.1) in the sense that (K ⎧ ⎪ ˜ ] + f1 (t, x, K ˜ 1 )K ˜ 1 − µ(t, x)M2 (K ˜ 1 ) ≤ 0, x ∈ (0, L), ⎪ L [K ⎪ ⎨ 1 1 ˜ 2 ] + f2 (t, x, K ˜ 2 )K ˜ 2 + A1 (x) µ(t, x)M2 (K ˜ 1 ) ≤ 0, x ∈ (0, L), L2 [ K A2 (x) ⎪ ⎪ ⎪ ⎩ Zi ˜ Z ˜ α1 Ki ≥ 0, α2 i K i ≥ 0, i = 1, 2.

(A.1)

Since system (3.1) is cooperative, the semiflow Φt : X → X is monotonic. (A.1) implies that Φt is point dissipative. Obviously, Φt : X → X is compact, ∀ t > 0. By [31, Theorem 3.4.8], it follows that Φt : X → X, t > 0, has a global compact attractor. We complete the proof.

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Appendix B. Proof of Lemma 3.4 Parts (i)-(ii). Assume that λ0 ≤ 0. By Lemma 3.2(i), we have limt→∞ Z1 (t, x) = 0, uniformly for x ∈ [0, L]. Thus, (3.1)(b), (3.1)(d) is asymptotic to system (3.5). By the theory of asymptotically periodic semiflows (see, e.g., [32] or [18, Section 3.2]) and Lemma 3.3, we see that the statements in (i) and (ii) are valid. Part (iii). Assume that λ0 > 0. By Lemma 3.2 (ii), we see that there is a unique positive ω-periodic ˜ 1 ] \ {0} such that Z1 (t, x) satisfies limt→∞ (Z1 (t, x) − Z ∗ (t, x)) = 0, uniformly for solution Z1∗ (t, ·) ∈ [0, K 1 x ∈ [0, L] provided that Z10 ̸≡ 0. Thus, (3.1)(b), (3.1)(d) is asymptotic to system (3.7). As mentioned before, system (3.7) admits a unique positive ω-periodic solution Z2∗ (t, x), which is globally asymptotically stable ˜ 2 ] \ {0}. Then it follows from the theory of asymptotically periodic semiflows for all initial conditions in [0, K (see, e.g., [32] or [18, Section 3.2]) that the statements in (iii) hold. Appendix C. Proof of Lemma 4.1 By [29, Theorem 1 and Remark 1.1] and the similar arguments in the proof of [30, Lemma 2.1], we can show that system (2.1) has a unique noncontinuable solution and the solutions to (2.1) remain nonnegative on their interval of existence if they are nonnegative initially. By a comparison argument, it is not hard to show that Y is positively invariant with respect to the periodic semiflow generated by system (2.1). It follows ˜ 2 that from Z2 (·, ·) ≤ K ⎧ ⎨ ∂F ˜ 2 , F ) + θ(t, x)F − m(t, x)F − η(t, x)F 2 , x ∈ (0, L), t > 0, ≤ Lf [F ] + c(t)H(t, x, K (C.1) ∂t ⎩F satisfies boundary conditions in (3.8)(b). In view of the assumption C(iii), it follows that there exists a Fˇ > 0 such that ˜ 2 , F ) + θ(t, x)F − m(t, x)F − η(t, x)F 2 ≤ 0, ∀ F ≥ Fˇ . c(t)H(t, x, K By comparison arguments, we deduce that F (t, x) is ultimately bounded. Then the semiflow Ψt : Y → Y is point dissipative. Obviously, Ψt : Y → Y is compact, ∀ t > 0. By [31, Theorem 3.4.8], it follows that Ψt : Y → Y, t > 0, has a global compact attractor. Appendix D. Proof of Lemma 4.2 (i). Since (Z1 (t, x), Z2 (t, x), F (t, x)) ≥ 0, we have ∂Z1 ≥ L[Z1 ](x) + f1 (t, x, Z1 )Z1 − µ(t, x)M2 (Z1 ), ∀x ∈ (0, L), t > 0. ∂t Assume that Z1 (t0 , ·) ̸≡ 0 at some t0 ⩾ 0. Consider the problem ⎧ ˆ ⎪ ⎪ ∂ Z1 = L[Zˆ1 ] + f1 (t, x, Zˆ1 )Zˆ1 − µ(t, x)M2 (Zˆ1 ), ∀x ∈ (0, L), t > t0 , ⎪ ⎨ ∂t ˆ ˆ Z ∂Z Z Z Z ∂Z ⎪α1 1 Zˆ1 (t, 0) − β1 1 ∂x1 (t, 0) = 0, α2 1 Zˆ1 (t, L) + β2 1 ∂x1 (t, L) = 0, t > t0 , ⎪ ⎪ ⎩ˆ Z1 (t0 , x) = Z1 (t0 , x), ∀x ∈ [0, L]. If Zˆ1 (t, x0 ) = 0 at some x0 ∈ [0, L] and t > t0 , then the strong maximum principle implies that x0 = 0 or L. However, the Hopf boundary lemma (see e.g., [33]) implies that Zˆ1,x (t, 0) > 0 (Zˆ1,x (t, L) < 0) if x0 = 0 (x0 = L) and 0 (L) is not a Dirichlet boundary point. A contradiction to the boundary conditions. Hence, Zˆ1 (t, x) > 0 for all x ∈ [0, L] other than the Dirichlet boundary points and t > t0 . By the comparison principle, we have Z1 (t, x) > 0 for t > t0 and all x ∈ [0, L] other than the Dirichlet boundary points.

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(ii). Since the solution semiflow Ψt : Y → Y of (2.1) is point dissipative, there is a constant Fˆ > 0 such that F (t, x) ≤ Fˆ for all x ∈ [0, L] and t ≥ t0 , and hence, A1 (x) ∂Z2 ≥ L2 [Z2 ] + f2 (t, x, Z2 )Z2 + µ(t, x)M2 (Z1 ) − σ(t, x)M1 (Z2 , Fˆ ) − H(t, x, Z2 , Fˆ ), ∂t A2 (x) for all x ∈ (0, L) and t ≥ t0 . By similar arguments as in (i), we can obtain that Z2 (t, x) > 0 for t > t0 and all x ∈ [0, L] other than the Dirichlet boundary points. (iii). The result can be similarly proved as in (i) by using the inequality ∂F ≥ Lf [F ] + θ(t, x)F − m(t, x)F − η(t, x)F 2 , x ∈ (0, L), t > 0. ∂t Appendix E. Proof of Theorem 4.1 Conditions in (i)-(iv) are those for all existing periodic solutions, including the trivial solution, to be ¯ (if exists), and E ˜ to be unstable in different situations. In particular, (i) consists of conditions for E0 , E ∗ , E ∗ ¯ (if exists) to be unstable (note that by Lemma 3.4(iii), unstable; (ii) consists of conditions for E0 , E , and E 0 ¯ ˜ and E ¯ to be unstable; (iv) consists when λ > 0, E is unstable if exists); (iii) consists of conditions for E0 , E, ¯ of conditions for E0 and E to be unstable. In the following, we will only give the detailed arguments for case ¯ (i). The cases (ii), (iii) and (iv) can be proved by similar arguments. For simplicity, we only assume that E ¯ does not exist. If E exists, the proof is similar. Denote the Poincar´e map of (2.1) as S = Ψω : Y → Y. Then S n = Ψnω , for n = 0, 1, 2, . . .. By Lemma 4.2, it follows that for any v0 ∈ W0 , we have Z1 (t, x, v0 ) > 0, Z2 (t, x, v0 ) > 0, and F (t, x, v0 ) > 0, for all t > 0 and x ∈ [0, L] except at the Dirichlet boundary points. In other words, Ψt (W0 ) ⊆ W0 , ∀ t ⩾ 0. It follows from Lemma 4.1 that S : Y → Y is point dissipative. Note that S n : Y → Y is compact for any n ∈ N. By [31, Theorem 3.4.8], S has a global compact attractor in Y. Let M∂ := {v0 ∈ ∂W0 : S n v0 ∈ ∂W0 , ∀ n ≥ 0}, and ω(v0 ) be the omega limit set of orbit O+ (v0 ) := {S n v0 : n ⩾ 0} for v0 ∈ Y. ˜ ·)}. Claim 1. ∪v0 ∈M∂ ω(v0 ) ⊆ {E0 (0, ·)} ∪ {E ∗ (0, ·)} ∪ {E(0, 0 n 0 Given v ∈ M∂ , we have S v ∈ ∂W0 , ∀ n ⩾ 0, that is, Z1 (nω, ·, v0 ) ≡ 0, or Z2 (nω, ·, v0 ) ≡ 0, or F (nω, ·, v0 ) ≡ 0, ∀ n ⩾ 0. Then one can further show that Z1 (t, ·, v0 ) ≡ 0, or Z2 (t, ·, v0 ) ≡ 0, or F (t, ·, v0 ) ≡ 0, ∀ t ⩾ 0. In the case where F (t, ·, v0 ) ≡ 0, ∀ t ⩾ 0, (Z1 , Z2 ) satisfies system (3.1). Since λ0 > 0, we see that limt→∞ (Z1 (t, x, v0 ), Z2 (t, x, v0 )) = (0, 0), or limt→∞ ((Z1 (t, x, v0 ), Z2 (t, x, v0 )) − (Z1∗ (t, x), Z2∗ (t, x))) = 0, uniformly for x ∈ [0, L] (see Lemma 3.4). This indicates that ω(v0 ) = E0 (0, ·) or E ∗ (0, ·). In the case where F (t, ·, v0 ) ̸≡ 0, for some t˜0 ⩾ 0, Lemma 4.2 implies that F (t, x, v0 ) > 0, for all t > t˜0 and x ∈ [0, L] except the Dirichlet boundary points. Then Z1 (t, ·, v0 ) ≡ 0, or Z2 (t, ·, v0 ) ≡ 0, for all t > t˜0 . If Z1 (t, ·, v0 ) ≡ 0, for all t > t˜0 , then it follows from the first equation in (2.1) that M1 (Z2 (t, ·, v0 ), F (t, ·, v0 )) ≡ 0, for all t > t˜0 . This implies that Z2 (t, ·, v0 ) ≡ 0, for all t > t˜0 . Hence, for all t > t˜0 , F satisfies (3.8). By Lemma 3.5, it follows that either limt→∞ F (t, ·, v0 ) = 0 or limt→∞ (F (t, x, v0 ) − F˜ (t, x)) = 0, uniformly for x ∈ [0, L]. This ˜ ·). If Z1 (t, ·, v0 ) ̸≡ 0, for some t˜1 > t˜0 , then one can further show that indicates that ω(v0 ) = E0 (0, ·) or E(0, Z1 (t, x, v0 ) > 0, for all t > t˜1 and x ∈ [0, L] except the Dirichlet boundary points. Thus, Z2 (t, ·, v0 ) ≡ 0, for all t > t˜1 . In view of the second equation in (2.1), we further see that Z1 (t, ·, v0 ) ≡ 0, for all t > t˜1 . It is a contradiction and this case cannot occur. From the above discussions, Claim 1 holds.

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Since λ0 > 0, there exists ρ > 0 such that λ(ρ) > 0, where λ(ρ) is the principal eigenvalue of ⎧ ⎪ ⎨φ1,t (t, x) = L1 [φ1 ] + (f1 (t, x, 0) − ρ)φ1 − µ(t, x)(M2,Z1 (0) + ρ)φ1 − λφ1 , x ∈ (0, L) φ1 satisfies boundary conditions in (3.1)(c), ⎪ ⎩ φ1 is ω -periodic in t.

19

(E.1)

Note that for the above fixed ρ, there exists δ > 0, such that when 0 < Z1 < δ, we have f1 (t, x, 0) − ρ < f1 (t, x, Z1 ) and M2 (Z1 ) < (M2,Z1 (0) + ρ)Z1 for all t ≥ 0 and x ∈ [0, L]. Since Ψt is continuous and Ψt (E0 ) = E0 , there exists ρ0 > 0 such that ∥Ψt (v)∥ < δ, for all t ∈ [0, ω] and v ∈ Y satisfying ∥v∥ < ρ0 . Claim 2. For the above ρ0 > 0, E0 is a uniform weak repeller of S for W0 in the sense that lim sup ∥S n (v0 ) − E0 ∥ ≥ ρ0 , ∀ v0 ∈ W0 . n→∞

˜ ∈ W0 such that lim supn→∞ ∥S n (˜ Suppose, by contradiction, there exists v v)−E0 ∥ = lim supn→∞ ∥S n (˜ v)∥ n < ρ0 . Then there exists n0 > 1 such that ∥S (˜ v)∥ = ∥Ψnω (˜ v)∥ < ρ0 for all n ≥ n0 . For any t ≥ n0 ω, there exists t0 ∈ [0, ω) and n ≥ n0 such that t = nω + t0 . Then ∥Ψt (˜ v)∥ = ∥Ψt0 (Ψnω (˜ v))∥ < δ. Hence, ˜ ) < δ for all t ≥ n0 ω and x ∈ [0, L]. It follows from the first equation of (2.1) that Z1 (t, x, v ⎧ ⎨ ∂Z1 ≥ L1 [Z1 ](x) + (f1 (t, x, 0) − ρ)Z1 − µ(t, x)(M2,Z1 (0) + ρ)Z1 , x ∈ (0, L), t ≥ n0 ω, (E.2) ∂t ⎩Z satisfies boundary conditions in (3.1)(c) for t ≥ n ω. 1

0

˜ ) ≫ 0, it follows that there exists m0 > 0 such that Z1 (n0 ω, ·, v ˜ ) ≥ m0 ϕρ (0, ·), where ϕρ ≫ 0 Since Z1 (n0 ω, ·, v λ(ρ)(t−n0 ω) ρ ˆ is the eigenfunction of (E.1) corresponding to λ(ρ). Note that Z1 (t, x) = m0 e ϕ (t − n0 ω, x) is the solution of ⎧ ˆ ∂ Z1 ⎪ ⎪ = L1 [Zˆ1 ](x) + (f1 (t, x, 0) − ρ)Zˆ1 − µ(t, x)(M2,Z1 (0) + ρ)Zˆ1 , x ∈ (0, L), t ≥ n0 ω, ⎨ ∂t (E.3) Zˆ1 satisfies boundary conditions in (3.1)(c) for t ≥ n0 ω, ⎪ ⎪ ⎩ˆ ρ Z1 (n0 ω, x) = m0 ϕ (0, x), x ∈ (0, L). Then the comparison principle implies that ˜ ) ≥ Zˆ1 (t, x) = m0 eλ(ρ)(t−n0 ω) ϕρ (t − n0 ω, x), ∀x ∈ [0, L], t ≥ n0 ω. Z1 (t, x, v ˜ ) is unbounded when t tends to ∞, which is a contradiction. Thus, Since λ(ρ) > 0, we see that Z1 (t, x, v Claim 2 is proved. Since Λ0 > 0, one can use the similar arguments in Claim 2 to show the following result: ˜ ·) is a uniform weak repeller of S for W0 in the sense that there exists a ρ˜0 > 0 such that Claim 3. E(0, ˜ ·)∥ ≥ ρ˜0 , ∀ v0 ∈ W0 . lim sup ∥S n (v0 ) − E(0, n→∞

Since ζ > 0 and ζ > ζ (see Remark 4.1), it follows that ζ ∗ > 0. Then there exists ρ¯ > 0 such that ζ(¯ ρ) > 0, where ζ(¯ ρ) is the principal eigenvalue of ⎧ ∗ ⎪ ⎨ϕt (t, x) = Lf [ϕ] + [c(t)(HF (t, x, Z2 , 0) − ρ¯) + θ(t, x) − m(t, x)]ϕ(t, x) − ζϕ(t, x), (E.4) ϕ satisfies boundary conditions in (3.8)(b), ⎪ ⎩ ϕ is ω -periodic in t. 0

∗

0

It follows from C(iii) that for the above fixed ρ¯, there exists δ¯ > 0 such that (HF (t, x, Z2∗ (t, x), 0) − ρ¯)F < ¯ Since H(t, x, Z2 , F ) for all t ≥ 0 and x ∈ [0, L] when ∥Z2 − Z2∗ (t, ·)∥ < δ¯ for all t ∈ [0, ω] and 0 < F < δ. ∗ ∗ ∗ ¯ Ψt is continuous and Ψt (E (0, ·)) = E (t, ·), there exists ρ1 > 0 such that ∥Ψt (v) − E (t, ·)∥ < δ, for all t ∈ [0, ω] and v ∈ Y satisfying ∥v − E ∗ (0, ·)∥ < ρ1 .

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Claim 4. For the above ρ1 > 0, E ∗ (0, ·) is a uniform weak repeller of S for W0 in the sense that lim sup ∥S n (v0 ) − E ∗ (0, ·)∥ ≥ ρ1 , ∀ v0 ∈ W0 . n→∞

Suppose, by contradiction, there exists v1 ∈ W0 such that lim supn→∞ ∥S n (v1 )−E ∗ (0, ·)∥ < ρ1 . Then there exists n1 > 1 such that ∥S n (v1 )−E ∗ (0, ·)∥ = ∥Ψnω (v1 )−E ∗ (0, ·)∥ < ρ1 for all n ≥ n1 . For any t ≥ n1 ω, there exists t1 ∈ [0, ω) and n ≥ n1 such that t = nω +t1 . Then ∥Ψt (v1 )−E ∗ (t, ·)∥ = ∥Ψt1 (Ψnω (v1 ))−E ∗ (t1 , ·)∥ < δ¯ for all t ≥ n1 ω. Hence, |Z2 (t, x, v1 ) − Z2∗ (t, x)| < δ¯ and F (t, x, v1 ) < δ¯ for all t ≥ n1 ω and x ∈ [0, L]. It follows from the third equation of (2.1) that ⎧ ⎨ ∂F ≥ Lf [F ] + c(t)(HF (t, x, Z2∗ , 0) − ρ¯)F + (θ(t, x) − m(t, x))F, x ∈ (0, L), t > n1 ω, (E.5) ∂t ⎩F satisfies boundary conditions in (3.8)(b) for t > n ω. 1

Since F (n1 ω, ·, v1 ) ≫ 0, it follows that there exists m1 > 0 such that F (n1 ω, ·, v1 ) ≥ m1 ϕρ¯(0, ·), where ¯ ¯ 1 ω) ϕρ ϕρ¯ ≫ 0 is the eigenfunction of (E.4) corresponding to ζ(¯ ρ). Note that Fˆ1 (t, x) = m1 eζ(ρ)(t−n (t − n1 ω, x) is the solution of ⎧ ˆ ∂F ⎪ ⎪ = Lf [Fˆ ] + c(t)(HF (t, x, Z2∗ , 0) − ρ¯)Fˆ + (θ(t, x) − m(t, x))Fˆ , x ∈ (0, L), t > n1 ω, ⎪ ⎪ ⎨ ∂t (E.6) ⎪Fˆ satisfies boundary conditions in (3.8)(b) for t > n1 ω, ⎪ ⎪ ⎪ ⎩ˆ F (n1 ω, x) = m1 ϕρ¯(0, x), x ∈ (0, L). Then the comparison principle implies that ¯ ¯ 1 ω) ϕρ F (t, x, v1 ) ≥ Fˆ (t, x) = m1 eζ(ρ)(t−n (t − n1 ω, x), ∀t ≥ n1 ω, x ∈ [0, L].

Since ζ(¯ ρ) > 0, F (t, x, v1 ) is unbounded when t tends to ∞. This is a contradiction, and hence, Claim 4 is proved. Define a continuous function p : Y → [0, ∞) by p(v0 ) := min{ min Z10 (x), min Z20 (x), min F 0 (x)}, x∈[0,L]

x∈[0,L]

x∈[0,L]

in the case of Robin boundary conditions in (2.2) or p(v0 ) := sup{β ∈ R+ : Z10 (x) ≥ β˜ e(x), Z20 (x) ≥ β˜ e(x), F 0 (x) ≥ β˜ e(x), ∀x ∈ [0, L]}, in the case of Dirichlet boundary conditions in (2.2), for all v0 = (Z10 , Z20 , F 0 ) ∈ Y, where e˜ is a given element ˜ + ). It follows from Lemma 4.2 that p−1 (0, ∞) ⊆ W0 and p has the property that if p(v0 ) > 0 or in Int(X 3 p(v0 ) = 0 with v0 ∈ W0 , then p(Ψt (v0 )) > 0, ∀ t > 0. That is, p is a generalized distance function for the periodic semiflow Ψt : Y → Y (see, e.g., [34]). ˜ ·)} From the above claims, it follows that any forward orbit of S in M∂ converges to {E0 (0, ·)} or {E(0, ∗ ∗ s ˜ or {E (0, ·)}; Further, {E0 (0, ·)}, {E(0, ·)} and {E (0, ·)} are isolated in Y, and that W ({E}) ∩ W0 = ∅, ˜ ·), E ∗ (0, ·), where W s ({E}) is the stable set of {E} with respect to S (see [34]). It for all E = E0 (0, ·), E(0, ˜ ·)} ∪ {E ∗ (0, ·)} forms a cycle in ∂W0 . By [34, Theorem 3], it is obvious that no subsets of {E0 (0, ·)} ∪ {E(0, follows that S is uniformly persistent with respect to (W0 , ∂W0 ) in the sense that there exists a ξ > 0 such that lim inf p(S n (v0 )) ≥ ξ, ∀ v0 ∈ W0 . n→∞

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Since S is uniform persistent and has a global attractor in Y, it follows from [35, Theorem 3.7] that S : W0 → W0 has a global attractor A0 . Let B0 = ∪t∈[0,ω] Ψt (A0 ). Then B0 is compact and B0 ⊂ W0 . By the continuity of p and compactness of B0 in W0 , we know that p(B0 ) admits a minimum value and ξ0 :=

1 min{p(u) : u ∈ B0 } > 0. 2

By the definition of B0 , we know that it attracts all solutions in W0 , that is limt→∞ d(Ψt (v0 ), B0 ) = 0 for any v0 ∈ W0 . This implies that limt→∞ p(Ψt (v0 )) ≥ ξ0 for any v0 ∈ W0 . Hence, (4.6) and (4.7) are true. Since S is compact and has a global attractor A0 in W0 , it follows from [35, Theorem 4.1] that S has a fixed point vp ∈ W0 , i.e., S(vp ) = vp . Therefore, system (2.1) admits a positive ω-periodic solution (Z1p (t, x), Z2p (t, x), F p (t, x)) = Ψt (vp )(x). The proof is completed. References [1] F. Casper, J.H. Thorp, Diel and lateral patterns of zooplankton distribution in the St. Lawrence River, River Res. Appl. 23 (2007) 73–85. [2] M.Z. Gliwicz, Predation and the evolution of vertical migration in zooplankton, Nature 320 (1986) 746–748. [3] C.J. Loose, P. Dawidowicz, Trade-offs in diel vertical migration by zooplankton: the costs of predator avoidance, Ecology 75 (8) (1994) 2255–2263. [4] P. Dawidowicz, J. Pijanowska, K. Ciechomski, Vertical migration of chaoborus larvae is induced by the presence of fish, Limnol. Oceanogr. 35 (1990) 1631–1637. [5] W.E. Neill, Induced vertical migration in copepods as a defence against vertebrate predation, Nature 345 (1990) 524–526. [6] I. A, McLaren, Demographic strategy of vertical migration by a marine copepod, Am. Nat. 108 (1974) 91–102. [7] M.D. Ohman, The demographic benefits of diel vertical migration by zooplankton, Ecol. Monogr. 60 (3) (1990) 257–281. [8] P.H.S. Picapedra, F.A. Lansac-Tˆ oha, A. Bialetzki, Diel vertical migration and spatial overlap between fish larvae and zooplankton in two tropical lakes, Brazil, Braz. J. Biol. 75 (2) (2015) 352–361. [9] Y. Iwasa, Vertical migration of zooplankton: A game between predator and prey, Amer. Nat. 120 (1982) 171–180. [10] G.C. Hays, A review of the adaptive significance and ecosystem consequences of zooplankton diel vertical migrations, Hydrobiologia 503 (2003) 163–170, In Migrations and Dispersal of Marine Organisms. Editors: M.B. Jones, A. Ing´ olfsson, ´ E. Olafsson, G.V. Helgason, K. Gunnarsson & J. Svavarsson. [11] W. Lampert, The adaptive significance of diel vertical migration of zooplankton, Funct. Ecol. 3 (1989) 21–27. [12] B.-P. Han, M. Stra˘skraba, Modeling patterns of zooplankton diel vertical migration, J. Plankton Res. 20 (1998) 1463–1487. [13] S.-H. Liu, S. Sun, B.-P. Han, Diel vertical migration of zooplankton following optimal food intake under predation, J. Plankton Res. 25 (2003) 1069–1077. [14] D. Bianchi, C. Stock, E.D. Galbraith, J.L. Sarmiento, Diel vertical migration: Ecological controls and impacts on the biological pump in a one-dimensional ocean model, Glob. Biogeochem. Cycles 27 (2013) 478–491. [15] S. Samanta, M. Alquran, J. Chattopadhyay, Existence and global stability of positive periodic solutionof tri-trophic food chain with middle predator migratory in nature, Appl. Math. Model. 39 (2015) 4285–4299. [16] Y. Zhang, F. Lutscher, F. Guichard, The effect of predator avoidance and travel time delay on the stability of predator–prey metacommunities, Theor. Ecol. 8 (2015) 273–283. [17] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, in: Pitman Search Notes in Mathematics Series, vol. 247, Longman Scientific Technical, Harlow, UK, 1991. [18] X.-Q. Zhao, Dynamical Systems in Population Biology, second ed., Springer, New York, 2017. [19] J. Fang, X. Yu, X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal. 272 (2017) 4222–4262. [20] Y. Jin, F.-B. Wang, Dynamics of a benthic-drift model for two competitive species, J. Math. Anal. Appl. 462 (2018) 840–860. [21] D. Kamykowski, J.H. Yamazaki, G.S. Janowitz, A Lagrangian model of phytoplankton photosynthetic response in the upper mixed layer, J. Plankton Res. 16 (1994) 1059–1069. [22] R.R. Hopcroft, J.C. Roff, Zooplankton growth rates: extraordinary production by the larvacean oikopleura dioica in tropical waters, J. Plankton Res. 17 (1995) 205–220. [23] D.C. Speirs, W.S.C. Gurney, Population persistence in rivers and estuaries, Ecology 82 (2001) 1219–1237. [24] P. Zhou, Y. Lou, Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations 259 (2015) 141–171. [25] S. Descloux, T. Datry, P. Marmonier, Benthic and hyporheic invertebrate assemblages along a gradient of increasing streambed colmation by fine sediment, Aquat. Sci. 75 (2013) 493–507. [26] D.B. Herbst, T.J. Bradley, A population model for the alkali fly at mono lake: depth distribution and chaning habitat availability, Hydrobiologia 267 (1993) 191–201. [27] G.F. Steward, J.P. Zehr, R. Jellison, J.P. Montoya, J.T. Hollibaugh, Vertical distribution of nitrogen-fixing phylotypes in a meromictic, hypersaline lake, Microb. Ecol. 47 (2004) 30–40.

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