Invasive-Invaded Interaction Incorporating a Bramson Model with Density-Dependent Diffusion and a Non-Lipschitz Reaction

The primary objective of the presented study is to investigate the pairwise interaction dynamics between invasive and invaded species, considering a model characterized by a non-regular, non-Lipschitz type reaction, as well as non-homogeneous diffusion. To achieve this, we begin with the foundational model proposed by Bramson in 1988 and tailor it to account for density-dependent diffusion and the non-Lipschitz type reaction, rendering it more applicable to our specific ecological scenario. Subsequently, our newly developed model is subjected to different analyses to ascertain the existence and uniqueness of positive weak solutions. It is noteworthy that density-dependent diffusive operators exhibit a property known as “finite propagation”, which manifests as the existence of a propagating front in the ecological system. Furthermore, we delve into the problem domain by employing the concept of traveling waves to identify specific solutions. A key outcome of our investigation is as follows: When both species propagate at significantly different speeds in the context of traveling waves, the interaction between them is deemed unstable, resulting in oscillations in the concentration of the invaded species. Conversely, when both species propagate within a similar range of speeds, the dynamics of the system are predominantly governed by the invasive species. AMS Subject Classification: 35K55, 35K57, 35K59, 35K65


Introduction
The analysis of reaction-diffusion models can be traced back to the seminal work of Kolmogorov, Petrovskii, and Piskunov (refer to [23]), as well as Fisher (as cited in [18]).In the latter case, reaction-diffusion models were applied to elucidate the dynamics inherent in gene interactions within the field of biology.In both of these foundational works, the authors introduced a novel class of solutions known as "traveling waves".This concept was pivotal in investigating evolutionary processes through the lens of classical parabolic operators.When such evolution occurs, species propagate through a given domain at a specific velocity, and the characteristics of the solutions are intimately tied to this propagating velocity.
The Fisher-KPP model has been a subject of extensive research, extending beyond its original context, to encompass diverse types of diffusion operators and various scopes and applications.For further insights, interested readers are directed to studies referenced in [5], [3], and [4].
Particularly noteworthy is the reformulation of the Fisher-KPP model using non-homogeneous diffusion operators, for which a general maximum principle is not readily available.These operators include the density-dependent p-Laplacian [6], higher-order diffusion [19], and nonlocal diffusion involving a fractional operator [12].
It is worth acknowledging that the rapid advancement of density-dependent diffusion parabolic operators in recent decades has opened up new avenues for comprehending reaction-diffusion problems.In studies such as [28] and [14], a nonlinear diffusive operator is introduced as a perturbation to a second-order Laplacian operator.Meanwhile, in [26] and [10], oscillatory profiles of solutions are characterized within an extended Fisher-Kolmogorov equation.This extension incorporates a higher-order operator, enabling the exploration of oscillatory spatial patterns in solutions, particularly relevant in materials science.
In our current analysis, our primary objective is to construct a model aimed at comprehending the dynamics of pairwise interactions between invasive and invaded species.To begin, it is pertinent to reference the definition of invasive species as per the Convention on Biological Diversity (refer to page 1, Chapter 1, [24]), which characterizes invasive species as those nonnative species that pose a threat to ecosystems, habitats, or other species.
There exist noteworthy classical analyses pertaining to the dynamics of invasive and invaded species.For instance, in the study conducted by Perum et al. [27], the authors investigated haptotactic motion in cells, leveraging an invasive model as the foundation for their research.Furthermore, the analysis presented in the work [20] delves into the study of traveling wave solutions and their stability, employing spectral techniques to gain insights into the formation of melanoma.
The dynamics associated with invasion can be effectively modeled through the lens of pairwise interaction.In this context, at a certain point in time, an invasive species, characterized by its concentration denoted as "u," infiltrates a region previously inhabited by the invaded species, marked by its concentration denoted as "v."This dynamic scenario was originally proposed by Bramson in his groundbreaking work [11], and it can be tailored to address the specific problem as follows: where d v and d v refer to the specific diffusion coefficients of each of the involved species, r v and r u are the specific rates of increase for each species and the coefficients given by α refer to the intra-specific and inter-specific competing balances.
The following lines provide a description about the main ideas considered to propose the new model discussed in this work: • The motion of invasive species in terms of diffusion is described using a generalized form of a diffusion operator.Specifically, several potential candidates have been considered for modeling in our case: 1) Porous Medium Equation: This is represented as ∇ • (u m−1 ∇u), where m > 1.It is a suitable choice for modeling invasive species diffusion.2) Higher-Order Operator: This is denoted as −(−∆) p w, with p ≥ 2. It offers an alternative approach for capturing the diffusion characteristics of invasive species.3) Thin Film Equation: Expressed as −∇ • (|w| n ∇∆u), where n > 0. This equation provides another framework to describe invasive species diffusion.4) p-Laplacian Operator: Represented as ∇ • (|∇w| p−2 ∇u), where p > 1.It is a valuable operator for modeling the diffusion of invasive species.Furthermore, our review of the existing literature in connection with density-dependent diffusion in biological applications has led us to consider the Keller and Segel model (as seen in [22]) as a reliable starting point for defining various types of density-dependent diffusion operators.In the Keller and Segel model, it is evident that there is local existence of a propagating front, which aligns with our research objectives.Other influential references exploring density-dependent diffusion include [1], [13], [30], and [29].These sources have contributed valuable insights into the application of density-dependent diffusion in biological contexts.• When the invasive species initiates its invasion of a given domain, there is a possibility that the invaded species is swiftly depleted or even annihilated.This phenomenon is more likely to occur at the outset of the invasion when the invasive species exerts particularly intense forces.However, it is important to note that the invaded species can exhibit resilience and may organize itself.In certain instances, this organization can lead to rapid recovery, marked by outbreaks in the concentration of the invaded species.These outbreaks in the concentration of the invaded species are assumed to be represented by a non-Lipschitz reaction term in the model.This term accounts for the dynamics whereby the invaded species, though initially suppressed, can experience surges in its population due to its ability to regroup and rebound.
Based on the ideas exposed, the following model is introduced: It is important to note that in our model, the reproducing coefficient denoted as r u > 0 signifies the renewal of invasive forces.The inter-specific term α uv quantifies the mortality rate of invasive agents due to interactions with the invaded species.
In the first equation within (2), we introduce a reproducing coefficient, denoted as r v , specifically for the invaded species.Additionally, we have intra-specific α vv and inter-specific α vu coefficients that account for the death rates of the invaded species.The term α vv can be associated with factors such as deaths due to hunger and disease within the invaded species, while α vu represents the mortality rate of the invaded species resulting from confrontations with the invasive species.
We define the problem domain to encompass the entire space, denoted as R d , where d > 1.In this context, we do not impose any boundary conditions.This choice allows us to investigate an open and unrestricted dynamics governed solely by the interactions between both species.
To initiate our analysis, we first examine the regularity and boundedness of solutions, which forms the foundation for the subsequent exploration of existence and uniqueness.The inclusion of density-dependent diffusion leads us to formulate the problem (2) within the framework of weak solutions.Consequently, we introduce a smooth test function denoted as Φ ∈ C ∞ (R d ).Subsequently, we delve into the profiles of solutions using a traveling wave formulation, which provides insights into specific characteristics of the proposed interaction.
The density dependant diffusion in (2) provides the following diffusivity functions: In the context of our model, where both parameters m and n are constrained to the interval (0, 1), it is worth noting that the problem is formulated with a fast diffusive term.Conversely, when both n and m exceed one, the diffusivity behaves in a manner characteristic of slow diffusion.In this latter scenario, a propagating front emerges, driven by diffusive processes (for a comprehensive discussion, refer to [16]).This front can be conceptualized as a geometric boundary within the domain.In the inner region, the invasion has effectively taken place, with the invasive species dominating.In contrast, the outer region is where the invasive species has yet to establish itself, and the invaded species prevails.The struggle between both species unfolds at the advancing front that propagates through the domain.Depending on the relative strengths of the species, the front may propagate in the direction of the dominant invasive species or the dominant invaded species.
It is important to emphasize that, in our analysis, we assume that both species maintain a slightly positive concentration at the front and are positive in regions sufficiently close to their advancing support.Furthermore, we posit that the positivity of solutions is preserved, at least locally, within the interval (0, T ], where T is chosen to be sufficiently small to uphold the parabolicity condition for the operator L. It is worth noting that, from a physical standpoint, we expect both solutions, denoted as u and v, to be positive at the moment of their pairwise interaction.Consequently, we draw upon the methodologies outlined in [7], [8], [9], and [32] to construct our analysis regarding the existence and uniqueness of solutions.
However, it is important to acknowledge that the presence of a non-Lipschitz reaction term necessitates the introduction of novel techniques to assess the impact of this reaction term on the regularity associated with the parabolic operator defined in Definition 1.
Supported by the aforementioned parabolicity property applicable to slightly positive solutions, we establish the following definition: Definition 2. (Lower and upper solutions).Consider the following set consisting in two upper solutions only: (v, û) ∈ C 2+µ, 1+µ/2 (R d × (0, T )).Additionally, assume the set formed of two lower solutions: (ṽ, ũ) ∈ C 2+µ, 1+µ/2 (R d × (0, T )).The set (v, û) formed by two upper solutions satisfies: Lv ≥ r v vq − α vv ṽq+1 − α vu ũṽ q , Lû ≥ r u û2 − α uv ṽ ũ. ( The set formed by (ṽ, ũ) is said to be a lower solution if: Consider, in addition, that where: and δ 0 shall be selected so that: Taking into account the parabolic nature of the operator L for solutions that are slightly positive, along with the continuity condition of the forcing terms for positive solutions, we can assert that both (v, û) and (ṽ, ũ) belong to L 1 (R d × (0, T )).For further elaboration and analysis, readers are directed to [21] and Chapter 3 of [31].
Let us consider a sequence of solutions denoted as (v (i) , u (i) ) i=0,1,2,... , which satisfy the following equations: The analysis concerning the regularity and existence of elements within the sequence (v (i) , u (i) ) i=0,1,2,... , as defined in (10), can be conducted using established techniques, given the mixed monotone behavior of each of the independent terms and the parabolicity of the diffusion operator L (refer to Chapter 3 of [31] and Theorems 2.1 and 2.2 in Chapter 7 of [25]).It is important to note that the equations presented in (10) are applicable to the lower and upper solutions as defined in Definition 2, and they hold for any initial distribution as provided in (8).Consequently, each element within the sequence complies with: In addition and following standard techniques, it is possible to show that the elements in the sequence are ordered (see the Ch. 12 in [25]).Therefore: Then, the sequence of upper solutions (v i) , û(i) ) is non-increasing and the sequence of lower solutions (ṽ (i) , ũ(i) ) is non-decreasing.

Definition 3. (Weak Formulation). Consider a smooth test function denoted as
), where 0 < ς < t < T .Then, within the time interval (ς, t], the following weak formulation applies: In these equations, a spatial ball centered at the origin with a radius J > 0 is considered.The test function is chosen such that ∂ β Φ(x0,t) ∂x β → 0 + for β ≥ 1 and for any point x 0 ∈ (∂J − , ∂J), where J 1 and 0 < < J/2.Now, consider any spatial point x 0 J and the interval R x = (x 0 , ∞).This interval R x defines a new problem that acts along the tail of the test function in R x × [0, t] with the following equations: with: Assume now that the following truncation holds for any 0 < α 1 < 1: where the function H may be u or v and p = {m, n}.The following truncation is defined for the non-Lipschitz reaction term: Based on the last defined truncated functions, the following problem (P Φ α1 ) is introduced accordingly: with the initial conditions stated in (16).
It is worth noting that due to the chosen form of truncation, the functions v α1 and u α1 remain bounded.Therefore, considering the smoothness and regularity of Φ, along with the parabolic nature of L, the monotonic behavior in the forcing terms, and the Lipschitz condition of V α , we can establish the existence of a single smooth test function Φ in R x = (x 0 , ∞) (for a comprehensive analysis in this regard, please refer to [25]).
The primary objective at this stage is to demonstrate that any weak solution, as defined in Definition 3, remains bounded.Furthermore, this boundedness persists when we revert to the non-Lipschitz condition by allowing α to approach zero in the expression (18).This outcome underscores the fact that, even in the presence of a non-Lipschitz reaction within the invaded species, the evolution follows a regular pattern for slightly positive solutions.In other words, the parabolic nature of the nonlinear operator is maintained throughout the evolution of solutions, ensuring regular behavior.
This regularity enables us to analyze the problem using an asymptotic approach in the tail of the test function and within the interval R x = (x 0 , ∞) for large values of x 0 1.
Theorem 1.Given the set of weak solutions (v, u) as per the expressions (13) and (14), such weak solutions are bounded for (x, t) ∈ R x × (ς, t].
Proof.Given any positive δ ∈ R, define a cut off function given by: In addition, it holds that: Consider now the weak formulations given in ( 13) and ( 14) for (x, t) ∈ R x × (ς, t] and a test function verifying Φ(ς) = Φ(t) = 0.Then, the following expressions are defined: where p = {m, n} and U can be either v or u, and is assumed to be arbitrary flat in R x .
Then, it holds that: To establish the boundedness of both v and u within the region R x ×(ς, t], we aim to demonstrate that the integrals on the right-hand side of the equations are bounded over their respective domains of integration.In order to achieve this, we rely on the following estimate (which can be found in [16]): Then, it holds that: Consider now the following form of a test function that complies with the conditions: where ς ≤ τ ≤ T .Note that η is selected for the convergence of the test function tail in R ∈ R x in accordance with the integral in (26), then: It suffices to take: so that the integral (26) is finite.
Coming now again to (24), it holds that: Here, we define m s as the maximum value of Φ over the interval (ς, T ].It is important to note that the solutions under consideration can be regarded as sufficiently flat, and both solutions can be interpreted as volumetric or mass concentrations, adhering to the constraints 0 < v < 1 and 0 < u < 1.
Due to the slightly positive nature of the solutions, we can allow α to approach zero in the expression for V α .Consequently, the non-Lipschitz condition does not have a significant impact on the convergence of the last integrals for any η > 0. In fact, the right-hand terms in these integrals are composed of constants.Thus, we can confidently assert the boundedness of (v, u) within the region R x × (ς, t].
By employing Definition 2 and establishing the boundedness of solutions, we can now demonstrate the convergence of the sequences of lower and upper solutions.
Proof.Consider the Definition 3 and assume a smooth test function as per (27), Φ ∈ C ∞ (R x ) in 0 < ς < t < T .The following weak formulations apply for (x, t) ∈ R x × (ς, t]: Rx where lim i→∞ α 1 (i) = 0. Supported by the dominated converge theorem, the following holds: For ṽ(i) : Making the subtraction between (34) and (35), the following holds: Given the required positivity in the solutions, the involved functions in the last expression are Lipschitz: Now, let us assume that we begin with a lower solution and an upper solution that are sufficiently close locally.In this scenario, suppose that û(i) converges to ũ(i) .Then, the expression (38) is applicable if v(i) also converges to ṽ(i) uniformly (with finite Lipschitz constants).Therefore: Since the upper solution is above the lower one, the above expression holds when v(i) converges to ṽ(i) , demonstrating the local convergence of solutions.
Similar assessments can be made for the u-equation, leading to an equivalent expression to that in (38).In this case, consider that the lower and the upper solutions for the species v are close, i.e., ṽ(i) → v(i) .Then, û(i) converges to ũ(i) .
Furthermore, the Dominated Convergence Theorem allows us to assert that the limiting functions maintain the same properties as each individual element in the sequence.As mentioned earlier, the regularity and positivity of the nonlinear operator L, which can be applied to each individual element in the sequence, enable us to establish a regularity statement for the limiting functions v and u: Building upon the previously obtained result regarding the convergence of upper and lower solutions, we introduce a new theorem to establish the uniqueness of weak solutions.Theorem 3. Consider the set of lower and positive solutions to (2), denoted by (v, u) > (0, 0) in R x × (ς, T ).This set of lower solutions locally coincides with the set of upper and positive solutions, represented by (v, û) > (0, 0).As a consequence, the solutions are unique.
Proof.Let us assume the existence of a set of upper solutions, denoted as (v, û), defined in R x × (ς, T ), which originates from the following initial data: where α 1 > 0 is a small positive value.Furthermore, the set (v, û) is determined by the following expressions: The set of lower solutions, that takes on from the functions (v(x, 0), u(x, 0)) = (v 0 (x), u 0 (x)), is provided as: Once again, let us consider a smooth test function Φ ∈ C ∞ (R x ) and formulate a weak formulation for the equations in question.Then, by subtracting the weak formulations of the upper and lower solutions, we can establish the following: consider now the continuous functions as follows: Consider a particular value of x, for τ = ς < T , the following applies: Now, assume that the following form of a smooth test function holds in 0 < ς < t < T and complying with Φ(ς) = Φ(t) = 0: that is defined for finite values of J 1 and κ 1 .Note that, the following holds: where ) and C 4 (κ 1 ) are two constants.For convenience, define the following functions: Based on the estimations exposed and returning to (43) and (44): The finite J 1 shall be chosen so that: It suffices to consider that: so that: The expressions given in ( 43) and (44) are then rewritten: The uniqueness of weak solutions is established locally.To illustrate this, consider that t ∈ (ς, ε 1 ) for 0 < ε 1 1.In such t-interval, we can define, at least by translation, a ε 1 -contact between the upper and the lower solutions for v, this means that: Let us make now the first time derivative in (58), then it holds that: Here, let us define M u = max t∈(ς,T ) {û}.The difference between the maximal and minimal solutions is positive and decreasing over time since both solutions should converge if uniqueness is satisfied.This implies that d(û−u) dt < 0. Now, consider A u = r u α u 2M u + ε 1 α uv .Through standard operations on the latter expression, we can establish the following inequality: where the minus sign in the exponential term is introduced so that d(û−u) dt < 0. Consider the expression (40), û(x, 0) − u(x, 0) = α 1 , then: The locally in time uniqueness is provided for α 1 → 0 + , then the following applies: The only compatible result in the previous expression is û = u, proving the local uniqueness of weak solutions.Return now to (57) and let us define the constant As previously exposed, make the first time derivative and solve by standard assessments, then it holds that: Making being the compatible result of the form: v = v, locally for any t ∈ (ς, T ).
The analysis presented allows us to assert the local uniqueness of solutions, as initially described in the theorem statement.

Analytical solutions for the propagating front
The objective is to derive analytical expressions for the propagating front where both species engage in a struggle.The invasive species seeks to conquer the domain, while the invaded species resists the invasion.This specific behavior in the pairwise interaction can be replicated by the nonlinear diffusion model we have considered.Notably, the porous medium diffusion, as described in (2), exhibits a property known as the finite propagation of species in a front.
It is essential to note that the analytical expression for the front is derived under the assumption of small values for each of the involved species, namely, v and u.This assumption accounts for the possibility of an initial confrontation when both species are still establishing themselves.Furthermore, this preliminary interaction defines the propagating support.
The problem we aim to analyze is then as follows: where |ϕ| 1 and compiles the higher order terms in the u−equation.The theorem to come provides the evolution, in the (x, t) variables, of the propagating front for the invaded species.Theorem 4. Considering the pairwise interaction, the following expression defines the evolution of the propagating front, primarily driven by the invaded species' ability to resist the invasion.It is important to note that the front is described in terms of the (x, t) variables: where and B v is a suitable constant to be determined.Additionally, v e represents the concentration levels of the invaded species required to contain the invasion within the propagating front.
Proof.First of all, consider the v−equation: The derivation of an analytical expression for the propagating front involves solving the above equation.To achieve this, we utilize a self-similar solution.It is widely recognized that the Porous Medium Equation allows for radially symmetric self-similar solutions (for more detailed information, refer to [31]).Therefore, a solution is expressed as follows: Replacing in (69) Balancing the leading terms in the last equation for the time exponents, it holds that: The results for ζ 1 and δ 1 are: Considering now the leading terms in (71), a selfsimilar profile of solution F is obtained by resolution of the following elliptic equation (note that r v = 1 for simplicity): The family of solutions to this equation is of the following form (see [31] and [15] for further insights): Replacing this form of solution in (74), and making standard operations, the following holds to determine A v and B v : Now, our aim is to derive an analytical expression for the propagating front when it is primarily driven by the invaded species, denoted as v.The ability of the invaded species to resist the invading force shapes the nature of the propagation and the evolution of the support.Let's assume that there exists a certain concentration of invaded species, denoted as v = v e , that resists within the front.Referring back to the self-similar solution, we have the following relationship: where F (η) is as per (75) and η = |x|t δ1 .Performing standard operations, the following expression is obtained for the propagating front: It is important to note that the constants in the last expression should be determined through observations, possibly relying on numerical simulations or laboratory experiments that pertain to the specific pairwise interaction being modeled.Recall that the constant A v is linked to the boundedness of the initial data and may represent the maximum concentration level in the initial distribution of the invaded species.A specific value for v e can also be obtained through numerical simulations of the relevant dynamics, which will be discussed in the following section.

Travelling waves
Certainly, the traveling wave profiles can be described as follows: , where n d ∈ R d represents a unitary vector, that defines the travelling wave direction of propagation and λ 1 , λ 2 refer to the propagation velocities, for each travelling wave solution.In addition, the profiles verify that: f, g : R → (0, ∞).In the sake of simplicity, assume that n d = (1, 0, ..., 0), then v(x, t) = f (ω 1 ), The problem ( 2) is then expressed based on the travelling wave profiles as follows (note that the derivatives are expressed with the standard notation, while is should be understood the derivation with regards to ω 1 for f and ω 2 for g): (79)

Numerical assessments
The numerical process aims to generate graphical solutions for the problem (79) to facilitate a deeper understanding of the dynamics.To achieve this, we will specify values for the constants presented in (2), specifically: q = 0.5; n = 2; m = 2; r v = 2; α vv = 1 α vu = 1; r u = 1; = 1; α uv = 1. (80) As a result, the only parameters we will modify in the numerical assessment are the traveling wave speeds.We performed numerical simulations using the bvp4c function in Matlab, which is wellsuited for solving boundary value problems (BVPs).The domain of integration was chosen to be sufficiently large to study the pairwise interaction without significant effects from pseudoboundary conditions required by the bvp4c function.Specifically, we used a domain size of R = 5000, although the provided figures only represent the main area of interaction between both species.We discretized the problem with 140,000 nodes and aimed for a required error of 10 −5 in each computed node.
The results of the numerical simulations are presented in Figures 1, 2, 3, 4, and 5 for different values of the traveling wave velocities.In these simulations, the invasive species originates from the right and interacts with the invaded species, resulting in the interaction front moving from right to left.Please note that traveling wave solutions are invariant under translation, so the invasive front occupies the entire domain.However, it is important to highlight that the existence of density-dependent diffusion coupled with a non-Lipschitz reaction can induce certain instabilities (oscillations) in the solutions, leading to values outside the natural interval [0, 1].In a real-world scenario, values beyond this interval are not physically meaningful.Therefore, if negative values are encountered, it should be understood as the species vanishing, and if concentrations exceed one, it should be interpreted as the species reaching its maximum concentration in the domain.

Discussions
The behavior of species propagating at significantly different speeds in the context of traveling waves, leading to unstable interactions and oscillations in the concentration of the invaded species, can be understood through the principles of dynamical systems.In systems where two species are interacting, such as an invasive species and a native species, the speed of propagation of each species' wavefront is a critical factor.This speed is determined by various biological and environmental factors such as reproductive rates, dispersal mechanisms, and the availability of resources.When there is a large disparity in the propagation speeds of the two species, the faster species tends to overtake the slower one, but this overtaking is not uniform or steady due to the complex interplay of other factors like spatial heterogeneity, resource competition, and environmental resistance.
As the faster species advances, it encounters pockets of resistance from the slower species or environmental factors, causing temporary setbacks or slowdowns.These irregularities lead to Figure 1.The plots depict traveling wave profiles, where the horizontal axis ω corresponds to either ω 1 or ω 2 .It is noteworthy to observe the oscillatory behavior of the invaded species in these profiles.This oscillation can be attributed to the density diffusive term, often referred to as the pressure density term in the porous medium operator, as well as the presence of a non-Lipschitz reaction.The invaded species exhibits a cycle of movement and resistance in its attempt to resist the invasion by the invasive species.This repetitive cycle of advancing and resisting leads to the observed oscillatory behavior.Eventually, the invasive species prevails, conquering the entire domain.Interestingly, there is a region in the profile where the invasive front appears less steep.This phenomenon appears to be associated with abrupt movements in the invaded species.Please note that the oscillations and dynamics in these profiles are a consequence of the complex interplay between the diffusion and reaction terms, which can lead to non-trivial and intricate behavior in the pairwise interaction between these two species.a non-uniform wavefront and result in oscillations in the concentration of the invaded species.Essentially, the invading species does not progress in a smooth, continuous front but rather in a series of surges and retreats, leading to a fluctuating population density of both species in the interaction zone.In our case, we observe that the invaded species is more sensitive than the invasive species and this is reflected by the observed oscillations.
On the other hand, when both species propagate within a similar range of speeds, the interaction dynamics change significantly.Since neither species has a distinct advantage in terms of propagation speed, the outcome of their interaction is more influenced by other factors, such as their respective reproductive rates, competitive abilities, and adaptability to the environment.In many cases, the invasive species, often having traits that give it a competitive edge (like higher reproductive rates or better resource utilization), dominates the interaction.The similar propagation speeds mean that the invasive species does not rapidly overrun the native species, but it steadily outcompetes it over time, leading to a scenario where the dynamics of the system are predominantly governed by the invasive species.
This described behavior is a reflection of the complexity inherent in ecological systems, where multiple factors interact in non-linear ways to determine the outcome of species interactions.
Figure 2.These graphs depict traveling wave profiles, where the horizontal axis ω corresponds to either ω 1 or ω 2 .In this case, we can observe a fast-paced and evidently unstable behavior in the invaded species.One notable feature is a rapid decline in the invaded species concentration, followed by a subsequent recovery as it continues to interact with the invasive species.Regardless of these fluctuations, it is apparent that the invasive species dominates when the traveling wave speeds are in the vicinity of 10.The observed rapid dynamics and instability in the invaded species are indicative of the complex and intricate nature of the pairwise interaction between these two species.The interaction dynamics can lead to non-trivial patterns and behaviors, as demonstrated in these profiles.
Figure 3.These graphs illustrate traveling wave profiles, with the horizontal axis ω corresponding to either ω 1 or ω 2 .The increase in traveling wave speeds results in a faster dynamic process.In these profiles, it becomes evident that the invasive species ultimately prevails over the invaded species.Notably, as the traveling wave speed increases, it enhances the competitive advantage of the invasive species, effectively reducing the time required for the invasion to be completed.The relationship between traveling wave speed and the outcome of the pairwise interaction highlights the significant impact of this parameter on the dynamics and outcome of the invasion process.As previously mentioned, the horizontal axis ω corresponds to either ω 1 or ω 2 .It is noteworthy that for higher and varying wave velocities, the dynamics exhibit oscillatory behavior.This behavior is particularly pronounced in the invaded species, which tends to recover periodically while maintaining the interaction.The interaction between the two species leads to a cyclic pattern of movement and resistance, resulting in the observed oscillations.This phenomenon underscores the intricate dynamics that can arise in pairwise interactions with density-dependent diffusion and non-Lipschitz reaction terms.

Conclusions
In this analysis, we introduced a novel extension to the classical Bramson model.Specifically, we incorporated a non-Lipschitz term into the equation for the invaded species to account for the potential emergence of sprouts of this species within the interaction domain.Our study began by establishing the regularity, existence, and uniqueness of weak solutions for the newly proposed equation.We derived an analytical expression for the propagating front in the invaded species, which reflected its ability to resist the invasive species' invasion.This expression was provided in terms of the (x, t) variables.
Subsequently, we examined the problem using a traveling wave formulation to gain deeper insights into the underlying dynamics.The traveling wave profiles associated with each of the involved species were computed using a numerical algorithm, and the results were visually presented.One significant conclusion drawn from our analysis is that when both species propagated at significantly different speeds, the interaction exhibited instability, leading to oscillations in the behavior of the invaded species.Conversely, when both species propagated within a similar range of speeds, the dynamics were predominantly governed by the invasive species.

Competing interests
The author declares no competing interest.

Availability of data and materials
This work has not associated data nor materials.

Figure 4 .
Figure 4.These graphs depict traveling wave profiles for substantially different values of traveling wave velocities.As before, the horizontal axis ω corresponds to either ω 1 or ω 2 .It is evident from these profiles that a decrease in the invasive species' speed leads to a corresponding decrease in the invasion dynamics.This is illustrated by the less steep graph in the invasive species profile when compared to the previous figures.The reduction in traveling wave velocity results in a slower invasion process, indicating that the speed of the invasive species plays a critical role in determining the dynamics of the pairwise interaction.

Figure 5 .
Figure 5.These graphs display traveling wave profiles for substantially different values of traveling wave speeds.As previously mentioned, the horizontal axis ω corresponds to either ω 1 or ω 2 .It is noteworthy that for higher and varying wave velocities, the dynamics exhibit oscillatory behavior.This behavior is particularly pronounced in the invaded species, which tends to recover periodically while maintaining the interaction.The interaction between the two species leads to a cyclic pattern of movement and resistance, resulting in the observed oscillations.This phenomenon underscores the intricate dynamics that can arise in pairwise interactions with density-dependent diffusion and non-Lipschitz reaction terms.