Existence, Uniqueness, and Input-to-State Stability of Ground State Stationary Strong Solution of a Single-Species Model via Mountain Pass Lemma

School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China Department of Mathematics, Chengdu Normal University, Chengdu 611130, China School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China Department of Mathematics, Sichuan Sanhe College of Professional, Luzhou 646200, China School of Information Science and Engineering, Chengdu University, Chengdu 610106, China


Introduction
e logistic system is one of the most classical models in ecology and mathematics, which is very important to the development of ecology [1].It is usually expressed as where x(t) represents the density or quantity of population x at time t, and r > 0 and K represent the intrinsic growth rate of population and environmental capacity, respectively.In 2011, Xiaoling Zou and Ke Wang investigated the long time behavior of the following stochastic ecosystem for single species ( [2], eorem 2): dx � x[a − bx]dt + αxdB(t). ( where a > 0 and b > 0 describe the growth rate and the intraspecific competition; α > 0 measures the intensity of the environmental disturbances.In recent years, model (2) has been widely adopted in many applications (e.g., [3][4][5]).A large number of facts have shown that the spatial scale and structure of the environment can affect population interaction [6,7] and community composition [8].In the landmark document [9], Kellam gave a large number of observations, which had a profound impact on the study of spatial ecology.First, he linked random walk with diffusion equation.e former is a description of the individual movement of some theoretical biological species, and the latter is a description of the density distribution of biological populations.He uses the data of muskrat transmission in Central Europe to prove that this connection is reasonable for small animals.Second, he combines diffusion with population dynamics and effectively introduces the reactiondiffusion equation into theoretical ecology.
In recent years, many dynamical systems, including reaction systems, have been considered as the theoretical branches of dynamical systems [10,11,12].In addition, the competition within the population is the participation of the adult population, and there is a period from infancy to adulthood.At the same time, this time-delay problem is affected by many stochastic factors such as weather, temperature, humidity, and so on.Besides, in real life, the factors that affect population growth do not change only at a fixed time but also occur randomly.When these factors occur, the system will also change randomly.As is well known, the phenomenon of population clustering is widespread in nature, which is likely to be completely affected by environmental factors and human factors.In this case, the growth curve of mosquitoes or small fish will be different from the previous form.is phenomenon can be expressed as a switch between two environmental states because the switching between different environments is not memory free; therefore, one can use continuous time Markov chain to model the situation of environment switching [13,14,15].
Inspired by some ideas and methods of related literature [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], particularly [17,18], we are to investigate the stability of stationary density of a single-species model with diffusion and delayed feedback under natural state.is study has the following highlights: (i) As far as we know, it is the first study to investigate the stability of stationary density of a single-species model with diffusion and delayed feedback under the Dirichlet zero boundary value.And the Dirichlet boundary value can well simulate the fact that the species lives in its biosphere, while the population density tends to zero at the boundary of biosphere due to the harsh condition.Different from exiting literature involved in Neumann zero boundary value which implies that there is no animal migration at the edge of the biosphere, the Dirichlet boundary value of this study admits the fact of animal migration, but no animals under study live on the border for a long time.(ii) It is the first comprehensive application of mountain pass lemma, variational technique, and the Lyapunov function method to derive the unique existence of globally exponentially stable positive stationary solution of a single-species model with diffusion and delayed feedback under the Dirichlet zero boundary value.(iii) e obtained stability criterion illuminates that under some suitable conditions, a certain internal competition is conducive to the overall stability of the population, and a certain amount of family planning is conducive to the overall stability of the population.(iv) Different from many existing literature related with the global stability of discontinuous systems [30][31][32] and time-delays reaction-diffusion systems [33,34], the weak stationary solution is regularized in this study.Most notably, the regularity technique of weak stationary solutions can also be applied to such existing literature [24,25,33,34] for the purpose of regularization of weak solutions.
In next sections, the authors have made the following arrangement: In Section 2, the authors present some descriptions on the ecosystem, and some necessary definitions and lemmas are presented.In Section 3, the authors utilize the existence technique and regularity method employed in [17] to derive the existence of positive strong stationary solution of the ecosystem.Moreover, utilizing the uniqueness technique used in [18,35] results in the unique existence of the stationary solution.Finally, the Lyapunov function method is applied to derive the stability criterion.In Section 4, numerical example and comparisons are given.And in final section, conclusions and further considerations are proposed.
For convenience, throughout of this study, we denote by λ 1 the first positive eigenvalue of Laplace operator − Δ in

System Descriptions
Denote by (Υ, F, P) the complete probability space with a natural filtration F t   t≥0 .Let S � 1, 2, . . ., n 0   and the random form process r(t): [0, +∞) ⟶ S { } be a homogeneous, finite-state Markovian process with right continuous trajectories with generator Π � (c ij ) n 0 ×n 0 and transition probability from mode i at time t to mode j at time t + δ, i, j ∈ S, where c ij ≥ 0 is the transition probability rate from i to j(j ≠ i) and Consider the following ecosystem with diffusion and delayed feedback, where Λ(u) is the external input, a > 0 and b > 0 describe the growth rate and the intraspecific competition, and Ω ∈ R 3 is the bounded domain with its boundary zΩ and is also a C 2,σ domain in R 3 (e.g., [17]).It is also suitable to the case that the species lives in two dimensional plane ([35], Remark 2.1).
Remark 1.Because only the adult is competitive for survival and there is a mature period from the larva to the adult, we consider the time-delayed system in this study, which is better to simulate this maturity problem.Particularly, the gain coefficient of time-delay feedback can be derived from the statistical data of the observed system.roughout this study, we assume (H1) the positive function Λ is only a microperturbation.at is, there exists a positive number ε > 0 small enough such that lim u⟶∞ Λ(u) where 2 * − 1 > θ � (θ 2 /θ 1 ) > 2 with θ 2 and θ 1 being a pair of coprime odd numbers.And Λ(•) is continuous and Λ(u) ≥ 0 for all u ≥ 0. Here, 2 * is the Sobolev critical exponent.In addition, Λ(u) � 0 for all u⩽0.
Remark 2. In essence, the restrictive condition θ � (θ 2 /θ 1 ) > 2 with θ 2 and θ 1 is set to ensure that the function u θ is odd, so that its primitive function is even.ereby, we can also assume in H1 that the function u θ is odd.
Let u * (x) be a stationary solution of system (4) implies that u * (x) is a solution of the following equation: Of course, each solution of equation ( 6) must be one of the solutions of system (4).

Definition 1.
e stationary solution u * (x) of system ( 4) is called the ground-state stationary solution of system (4) if u * (x) is the ground-state solution of equation ( 6).Definition 2. A solution u * (x) of equation ( 6) is called the strong solution of equation ( 6 To prove the main result of this study, we need the following lemmas (e.g., [17,20]): Lemma 1.Consider the following equation: where Ω is a C k+2,α domain of R n , and f satisfies the following conditions: (a) ere exists 0 < r ≤ 1 such that for any given positive number M, en, the solution of equation ( 6) in H 1 0 (Ω) is the strong solution.In addition, u ∈ C k+2,δ for δ � αr k+1 .
Remark 3. Lemma 3 is the mountain pass lemma without the PS condition (e.g., [17]).If, in addition, Ψ satisfies the PS condition, then c * is a critical value of Ψ. Besides, let Ψ be the functional corresponding to equation ( 6), then u * (x) must be a ground state solution of equation ( 6) if u * (x) is a critical point of the functional Ψ with Ψ(u * (x)) � c * defined in (10) of Lemma 3.

Main Result
First, we may present the existence of a stationary strong solution u * (x) of system 2.1.In addition, it is necessary to guarantee that u * (x) ≥ 0 and u * (x) ≠ 0, which may be proved as follows: Theorem 1. Suppose the condition H1 holds, and if en, there is a ground-state strong stationary solution for system (4).
Proof.Let u * (x) be a positive stationary solution of system (4), satisfying where μ > 0 is a constant, and , and a critical point of the functional Ψ is corresponding to the solution of equation (12).Next, we claim that Ψ satisfies the condition of the mountain road geometry.In fact, obviously Ψ(0) � 0.
e microperturbation condition (H1) yields that there are there positive constants c 0 , m 1 , m 2 with or which implies and then, Moreover, Next, ( 13) and ( 18), Poincare inequality and Sobolev embedding theorem yield that there is a positive constant c 1 > 0 such that Besides, (11) and small ε > 0 lead to which together with θ > 2 means that we can set On the other hand, it follows by (15) and m 1 < 1 that which implies or We may select u ∈ H 1 0 (Ω) with u ≥ 0, and then, Let φ 1 (x) > 0 with ‖φ‖ � 1 be the eigenfunction corresponding to the first positive eigenvalue λ 1 (e.g., [18]), and set u � sφ; then, Ψ(sφ) ⟶ − ∞ if s ⟶ + ∞, so that there exists s 0 > 0 satisfying ‖s 0 φ‖ ≥ ρ and Ψ(sφ) < 0. And then, Ψ satisfies the condition of the mountain road geometry.According to mountain pass lemma, let Γ be the set of all paths connecting 0 and e � s 0 φ. at is, Set en, c * ≥ α, and Ψ possesses a critical sequence on c * , say u k   ⊂ H 1 0 (Ω) with Ψ(u k ) ⟶ c * and Ψ ′ (u k ) ⟶ 0 in (H 1 0 (Ω)) * .at is, for any given ϵ > 0, there exists k big enough such that and where o(1) represents such an infinitesimal that o(1) ⟶ 0 when k ⟶ ∞.
Indeed, u * (x) ≠ 0 is the nonnegative solution of the following Dirichlet problem: where It is easy from the assumptions on Λ to verify that f satisfies the conditions (a)-(c); then, Lemma 1 yields that u * (x) is the strong solution.

Complexity
Set v(t, x) � u(t, x) − u * (x).Since u * (x) is a stationary solution of system (4), system (4) is equivalent to the following system: where Obviously, u * (x) of system ( 4) is corresponding to the zero solution of system (38).
Equip system (38) with the initial value, Moreover, we give some suitable assumptions as follows: H2. ere are positive numbers M 0 , N 0 , such that H3. ere is a positive number M 1 > 0, such that □ Remark 4. Everyone knows the fact that the population density of any species must have the bounded below or the species will die out.For example, when the population density of whales is lower than a certain degree, it will be difficult for male and female whales to meet each other in the vast sea, leading to the extinction of the species.Besides, due to the limited resource, the population density of any species must have supper boundedness.So, the condition H2 is a suitable assumption.ere are some techniques on the existence and uniqueness of positive stationary solution in the proofs ( [18], eorems 1-2), which are also employed in the proofs of ( [35], eorems 1-2).But the methods used in the proof of eorem 1 in this study are different from those of both [18,35].Besides, we are willing to consider similarly the uniqueness of the positive stationary solution in this study.

Theorem 2. If all the conditions of eorem 1 hold, and if, in addition
then u * (x) is the unique stationary solution of system (11).
Proof.Let u, w both are the stationary solutions of system (4).First, the condition H2 yields Since u, w both are the stationary solutions of system (4), then Poincare inequality yields which proves u � w, and the proof is completed.

□
Theorem 3. Suppose the conditions H1-H3 and ( 45) hold and if there are positive numbers p r (r ∈ S), k 1 > 0 such that then the null solution of system (38) with the initial value (40) is the globally exponential input-to-state stability; at the same time, u * (x) is the globally exponential input-to-state stability at the convergence rate (λ/2), where α � min r∈S (2λ ), and λ is the unique positive solution of λ � α − βe λτ .

□
Remark 5.In this study, we employ mountain pass lemma and variational technique to derive the existence of positive stationary solution, which is different from the methods in [18].Particularly, ground-state solution is more suitable to practical engineering (e.g., [36][37][38][39][40]).Besides, the equilibrium points of ecosystems with the Neumann boundary value are always constants solutions, while equilibrium points of the system (4) are always the nontrivial solutions of the corresponding elliptic equation, which need the existence criterion of the solutions for the elliptic equation.And it adds the computation complexity of the results obtained in this study.Also, our model and method are different from those in [41][42][43][44][45][46].
In Example 1, replacing ε � 0.00001 with ε � 0.0001 and other data unchanged, direct computation yields the convergence rate (λ/2) � 21.59%.Remark 6.Table 1 provides that under some suitable conditions, the smaller the external input disturbance, the faster the stability of the natural ecosystem (Figure 1).
In example 1, replacing b � 0.1 with b � 0.2 and other data unchanged, direct computation yields the convergence rate (λ/2) � 28.98%.Remark 7. Table 2 provides that a certain degree of inhibition and competition within the population is beneficial to the overall stability of the population for the natural ecosystem of a single-species model.Figure 2 shows that the bigger the intrapopulation competition intensities, the faster the stability of a single-species system.8 Complexity Remark 8. Table 3 provides that due to the loss of natural enemies in a single-species model, the higher the natural population growth rate, the slower the stability of the ecosystem (Figure 3).

Remark 9.
Different from [2,3], the population density boundedness of the species was considered in this study due to the important factor (Remark 4 for details).In Example 1, authors assume N 0 � 2 and M 0 � 10, and we can see it from Figures 1-3 that 2 < u < 10.

Conclusions
In this study, critical point theory and variational methods are utilized to derive the unique existence of stationary solution of the single -species model, which is positive and strong.Moreover, the geometric characteristic of saddle point in the mountain pass lemma guarantees that the positive strong stationary solution is the ground state one.Moreover, the method of Lyapunov function yields the global exponential stability of the ground-state classical positive stationary solution which is the unique stationary solution of the ecosystem.Besides, impulse control reflects the human intervention in the natural ecosystem (e.g., [35], eorem 3); we may consider the next study on impulsive stabilization of the single-species ecosystem of this study.In addition, we propose mathematical conjectures that ( [18], Problem 4) and ([18], Problem 1) may be correct even in the case of the stochastic differential system.In this study, we propose the mathematical conjecture that under suitable conditions, small diffusions that can make the unique stable equilibrium point of the delayed ordinary differential system become multiple stationary solutions of its corresponding partial differential system, even in the case of the stochastic model.Moreover, how to give a global stability invariance criterion of a stochastic model similar as ([18], Corollary 3.4)? is is an interesting problem.Finally, we may consider an interesting application to epidemic control (e.g., [19]). is study is involved in a single-species biological dynamic system.If we investigate the dynamics of a single-species infectious disease model, it is another interesting problem, for the human society may approximately be regarded as a singlespecies model.Complexity

Table 1 :
In Example 1, replacing a � 0.2 with a � 0.23 and other data unchanged, direct computation yields the convergence rate (λ/2) � 19.08%.Comparisons the influences on the convergence rate (λ/2) under different perturbations with the same other data.

Table 3 :
Comparisons the influences on the convergence rate (λ/2) under different growth rates with the same other data.