#### Abstract

In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and the Lyapunov function method to derive the unique existence of the positive classical stationary solution of a single-species ecosystem. Particularly, the geometric characteristic of saddle point in the mountain pass lemma guarantees that the equilibrium point is the ground state stationary solution of the ecosystem. Based on the obtained uniqueness result, the authors use the Lyapunov function method to derive the globally exponential stability criterion, which illuminates that under some suitable conditions, a certain internal competition is conducive to the global stability of the population, and a certain amount of family planning is conducive to the overall stability of the population. Most notably, the regularity technique of weak stationary solution employed in this study can also be applied to some existing literature related with time-delays reaction-diffusion systems for the purpose of regularization of weak solutions. Finally, an illuminative numerical example shows the effectiveness of the proposed methods.

#### 1. Introduction

The 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 aswhere represents the density or quantity of population at time , and and 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], Theorem 2):where and describe the growth rate and the intraspecific competition; measures the intensity of the environmental disturbances. In recent years, model (2) has been widely adopted in many applications (e.g., [3–5]). 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. The 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 reaction-diffusion 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. This 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 [16–31], 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. This 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)The 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 [30–32] 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 the first positive eigenvalue of Laplace operator in . Denote . Denote by , the norm of , and by , the first positive eigenvalue of Laplace operator in . Besides, we denote for and for matrix .

#### 2. System Descriptions

Denote by the complete probability space with a natural filtration . Let and the random form process be a homogeneous, finite-state Markovian process with right continuous trajectories with generator and transition probability from mode at time to mode at time ,where is the transition probability rate from to and , and .

Consider the following ecosystem with diffusion and delayed feedback,where is the external input, and describe the growth rate and the intraspecific competition, and is the bounded domain with its boundary and is also a domain in (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.

Throughout this study, we assume (H1) the positive function is only a microperturbation. That is, there exists a positive number small enough such thatwhere with and being a pair of coprime odd numbers. And is continuous and for all . Here, is the Sobolev critical exponent. In addition, for all .

*Remark 2. *In essence, the restrictive condition with and is set to ensure that the function is odd, so that its primitive function is even. Thereby, we can also assume in H1 that the function is odd.

Let be a stationary solution of system (4) implies that 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. *The stationary solution of system (4) is called the ground-state stationary solution of system (4) if is the ground-state solution of equation (6).

*Definition 2. *A solution of equation (6) is called the strong solution of equation (6) if .

To prove the main result of this study, we need the following lemmas (e.g., [17, 20]):

Lemma 1. *Consider the following equation:where is a domain of , and satisfies the following conditions:*(a)*There exists such that for any given positive number ,*(b)*If and , or and , then (as ) holds uniformly for *(c)*.*

Then, the solution of equation (6) in is the strong solution. In addition, for .

Lemma 2. *Let . Then, there is a conclusion that . Besides, if , if , if , and if . In addition, if and if . Besides, if .*

Lemma 3 (Mountain pass lemma without the PS condition). *Let be a Banach space, , satisfying , and there exists such that . Besides, there is , such that . Let be the set of all paths connecting 0 and , that is,**Set*

Then, , and possesses a critical sequence on .

*Remark 3. *Lemma 3 is the mountain pass lemma without the PS condition (e.g., [17]). If, in addition, satisfies the PS condition, then is a critical value of . Besides, let be the functional corresponding to equation (6), then must be a ground state solution of equation (6) if is a critical point of the functional with defined in (10) of Lemma 3.

#### 3. Main Result

First, we may present the existence of a stationary strong solution of system 2.1. In addition, it is necessary to guarantee that and , which may be proved as follows:

Theorem 1. *Suppose the condition H1 holds, and if*

Then, there is a ground-state strong stationary solution for system (4).

*Proof. *Let be a positive stationary solution of system (4), satisfyingwhose functional iswhere is a constant, and . It is obvious that , and a critical point of the functional is corresponding to the solution of equation (12). Next, we claim that satisfies the condition of the mountain road geometry. In fact, obviously .

The microperturbation condition (H1) yields that there are there positive constants with such thatorwhich impliesand then,Moreover,Next, (13) and (18), Poincare inequality and Sobolev embedding theorem yield that there is a positive constant such thatBesides, (11) and small lead towhich together with means that we can setsuch that .

On the other hand, it follows by (15) and thatwhich impliesorWe may select with , and then,Let with be the eigenfunction corresponding to the first positive eigenvalue (e.g., [18]), and set ; then, if , so that there exists satisfying and . And then, satisfies the condition of the mountain road geometry. According to mountain pass lemma, let be the set of all paths connecting 0 and . That is,SetThen, , and possesses a critical sequence on , say with and in . That is, for any given , there exists big enough such thatandwhere represents such an infinitesimal that when .

(17) yieldsSimilar as the methods of [17], (24)–(27), employing (29)–(33) results inwhere both are the positive constants, and due to the small , which means the boundedness of . Due to , equation (12) is the subcritical growth. It is a routine proof of the fact that is sequently compact, say in and , which implies . Besides, is the critical point of , so thatIn (35), setting , Lemma 2 leads towhich implies that a.e. . Now, we have proved that and .

Similar as that of [17], now we claim that the abovementioned is the strong solution.

Indeed, is the nonnegative solution of the following Dirichlet problem:whereIt is easy from the assumptions on to verify that satisfies the conditions (a)–(c); then, Lemma 1 yields that is the strong solution.

Set . Since is a stationary solution of system (4), system (4) is equivalent to the following system:whereObviously, 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. There are positive numbers , such thatH3. There is a positive number , 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.

There are some techniques on the existence and uniqueness of positive stationary solution in the proofs ([18], Theorems 1-2), which are also employed in the proofs of ([35], Theorems 1-2). But the methods used in the proof of Theorem 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 Theorem 1 hold, and if, in addition,then is the unique stationary solution of system (11).*

*Proof. *Let both are the stationary solutions of system (4). First, the condition H2 yieldsSince both are the stationary solutions of system (4), then Poincare inequality yieldswhich proves , and the proof is completed.

Theorem 3. *Suppose the conditions H1–H3 and (45) hold and if there are positive numbers , such thatthen the null solution of system (38) with the initial value (40) is the globally exponential input-to-state stability; at the same time, is the globally exponential input-to-state stability at the convergence rate , where , , and is the unique positive solution of .*

*Proof. *Consider the following Lyapunov function:where is a positive number for each .

First, H2 yieldsH3 yieldsLet be the weak infinitesimal operator (e.g., [21]), such thatwhere .

Now, ([36], Lemma 2) yields thatwhere , This completes the proof.

*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., [36–40]). 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 [41–46].

#### 4. Numerical Example

*Example 1. *In system (4), let , then ([18], Remark 14). , then , and condition (11) holds. Setwhere , then H1 holds. Let , then direct computation yields , and obviously, (45) holds. Then, Theorems 1-2 yield is the unique stationary solution of system (4), and is the ground state stationary strong solution of system (4).

Moreover, set , and , , and direct computation yields (46) holds for , and . According to Theorem 3, the null solution of system (38) with the initial value (40) is globally exponential input-to-state stability; at the same time, is globally exponential input-to-state stability at the convergence rate .

In Example 1, replacing with and other data unchanged, direct computation yields the convergence rate .

*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 with and other data unchanged, direct computation yields the convergence rate .

*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.

In Example 1, replacing with and other data unchanged, direct computation yields the convergence rate .

*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 and , and we can see it from Figures 1–3 that .

#### 5. 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], Theorem 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)? This is an interesting problem. Finally, we may consider an interesting application to epidemic control (e.g., [19]). This 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 single-species model.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

Ruofeng Rao wrote the article. Quanxin Zhu and Huang Jialin guided the writing of the article and gave their contributions on the revision, being the corresponding authors in charge of the communication of this article. All the authors typed, read, and approved the article.

#### Acknowledgments

The authors would like to thank the National Natural Science Foundation of China (61773217) and Chinese Science and Technology Department of Sichuan Province for funding this work through the application basic research project (2020YJ0434) and thank Chengdu Normal University for funding this work through major scientific research projects of Chengdu Normal University (CS19ZDZ01).