Biological invasion is the process by which organisms move from their original habitat to a new environment, either naturally or through human intervention. The proliferation of biological invasion involves a complex chain of events, including invasion, diffusion, settlement, adaptation, and other stages. Each of these stages is necessary for the success of the invasion. Due to the complexity of the problem of species invasion, [1, 2] attempts have been made to establish a unified biological framework to describe this issue. However, there are still many shortcomings, so it is necessary to establish a biological invasion model that is more in line with the actual situation.

As the issue of biological invasion gains importance in ecology, it is crucial to establish a mathematical model to describe it. Various mathematical models have been used to describe population growth in biology, such as Volterra integral equations [3, 4] and Logistic models [5, 6] to describe single race growth or intermediate competition. Additionally, the free boundary model has been used to describe tumor growth and biological invasion. In addition, the processing of free boundary is also a difficult problem. In order to better deal with this problem [7] use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. To investigate regional separation in population ecology, [8, 9] studied the free boundary model with local diffusion. They demonstrated the existence and uniqueness of the global solution for the model. The population expansion process was further studied by [10], who also improved the derivation of free boundary conditions. [11-13] describe a biological invasion model with free boundaries, which reduces the complexity of the problem and better simulates actual activities compared to fixed boundaries. [14] presents a numerical solution of nonlinear partial differential equations to characterize the spatial flow of a general animal population model. [15] used the He-Laplace method combined with fractional order complex transformation (FCT) to study a two-scale population growth model in a closed system.

The red fire ant (Solenopsis invicta Buren) has the potential to cause significant damage to local biodiversity, agriculture and public health. Consequently, it is vital to implement an effective eradication strategy that involves thorough monitoring and the destruction of the ant at the earliest stage of the invasion in order to prevent its expansion [16]. The distribution and occurrence of red fire ants in Yunnan were analyzed in [17]. [18] presents the main parameters of several red fire ant invasion events in South China, obtained through field investigation. The invasion history and propagation rules of red fire ants were studied by establishing and applying relevant models.

The population growth of invasive species can be modelled using the logistic growth equation, which is an ordinary differential equation (ODE):

$ \begin{aligned} u'(t)=au-du-bu^2 \end{aligned} $

The population of the invasive species is represented by u, while a and d represent the birth and natural death rates, respectively. Additionally, b represents the death rate due to intra-species competition.

The diffusive logistic equation in the $ {{\mathbb{R}}^{n}} $ as follow:

$ \begin{aligned} {{u}_{t}}-d\Delta u=u(a-bu), t>0, x\in {{\mathbb{R}}^{n}} \end{aligned} $

Here $ d\Delta u $ represents the local dispersal of species. However, scientific research has shown that local diffusion operators cannot accurately describe nonlocal actions in space. Nonlocal diffusion equations are better suited to describe the diffusion of new species in new habitats. These phenomena are well described by nonlocal diffusion equations:

$ \begin{aligned} {{u}_{t}}=\Delta u+u(1-J*u), t>0, x\in \mathbb{R} \end{aligned} $

Here $ J(x)={{J}_{\delta }}(x)=\frac{1}{\delta }J(\frac{x}{\delta }) $ is kernel function and satisfies threshold condition,

$ \begin{aligned} J(x)\in {{L}^{1}}(\mathbb{R}), J(x)\ge 0, \int_{R}{J(x)dx=1}, J*u=\int_{R}{J(x-y)u(y)}dy \end{aligned} $

Here $ \delta >0 $ is used to represent the strength of the non-local effect, and the non-linear vector $ J*u $ represents the competition for space resources between individuals within a race or with individuals of other species.

[3] proposed a model that assumes that a single population has no interaction with the outside world.

$ \begin{aligned} \left\{ \begin{matrix} {{u}_{t}}=\Delta u+au-b{{u}^{2}}-uJ*u & in\text{ }D=(0, \infty )\times \Omega \\ \frac{\partial u}{\partial n}=0 & \text{ }on\text{ }\Gamma =(0, \infty )\times \partial \Omega \\ u(0, x)={{u}_{0}}(x) & for\text{ }x\in \bar{\Omega } \\ \end{matrix} \right. \end{aligned} $

Here $ J*u=\int_{0}^{t}{J(t-s)u(s, x)}ds $, $ (t, x)\in D $ is the genetic impact of the past on the present, denotes a bounded domain in a new habitats $ {{\mathbb{R}}^{n}} $ with boundary $ \partial \Omega $. $ \partial /\partial n $ represents the exterior normal derivative to $ \partial \Omega $.

[6] proposed a free boundary model based on the following diffusive logistic problem and a unique global solution is proved:

$ \begin{aligned} \left\{\begin{matrix} {{u}_{t}}=d\Delta u+u(a-bu) & t>0, 0<x<h(t) \\ {{u}_{x}}(t, 0)=u(t, h(t))=0 & t>0 \\ {h}'(t)=-\mu {{u}_{x}}(t, h(t)) & t>0 \\ h(0)={{h}_{0}}, u(0, x)={{u}_{0}}(x) & 0<x<{{h}_{0}} \\ \end{matrix} \right. \end{aligned} $

Here $ x=h(t) $ is the spreading front of invasive species, a represents the intrinsic growth rate of the species, b is internal competition rate, d is dispersal rate. The long-term dynamic behavior of the model is either vanishing or diffusion.

Established according to the above equation, we can get:

$ \begin{equation} \left\{\begin{matrix} {{u}_{t}}=d{{u}_{xx}}+u(a-bu-({{\phi }^{*}}u)) & t>0, 0<x<h(t) \\ {{u}_{x}}(t, 0)=u(t, h(t))=0 & t>0 \\ {h}'(t)=-\mu {{u}_{x}}(t, h(t)) & t>0 \\ h(0)={{h}_{0}}, u(0, x)={{u}_{0}}(x) & 0<x<{{h}_{0}} \\ \end{matrix} \right. \end{equation} $

(2.1)

Here $ x=h(t) $ is moving boundary to be determined and it denotes the spreading front, $ u(t, x) $ represents the spreading of a new species in habitat, a, b, r, d and $ \mu $ are given positive constants. The parameter a represents the intrinsic growth rate of the species, this is the maximum rate of growth a species can have when food and space are abundant, b measures its intraspecific competition, and d is the dispersal rate. $ \mu>0 $ represents the ability of species to expand at boundaries. The free boundary $ h'(t)=-\mu {{u}_{x}}(t, h(t)) $ is a special case of Stefan condition. The integral $ ({\phi }^{*}u)(t, x)=\int_{{t}_{0}}^{t} {\phi }^{*}u(t-\tau)u(\tau) {\rm d}\tau $ is a hereditary term containing the effect of the past history on the present growth rate, $ [0, {h}_{0}] $ represents species initial living space.

The initial function $ {{u}_{0}(x)} $ satisfies

$ \begin{eqnarray} u_{0}(x)\in C^{2}([0, h_{0}]), u_{0}(h_{0})=u_{0}^{'}(0)=0\ and \ u_{0}>0 \ in\ [0, h_{0}) \end{eqnarray} $

(2.2)

and $ {\phi }^{*}u $ satisfies

$ \begin{eqnarray} \phi \in C(\mathbb{R}) \cap \mathbb{L}^{1}(\mathbb{R}), \phi (x)= \phi (-x) \ and\ \int_{R} \phi(x) {\rm d} x=1. \end{eqnarray} $

(2.3)

Now we prove the existence and uniqueness of the local solution of problem (2.1) by using the principle of compression mapping.

Theorem 3.1 For any given $ {{u}_{0}} $ satisfying (2.2) and any $ 0<\alpha<1 $, there is a $ T>0 $ such that problem (2.1) admits a unique solution

$ \begin{aligned} (u, h)\in C^{(1+\alpha)/2, 1+\alpha}(D_{T}) \times C^{(1+\alpha)}([0, T]). \end{aligned} $

Moreover, $ \| u\|_{ C^{(1+\alpha)/2, 1+\alpha}(D_{T})} + \| h\|_{C^{(1+\alpha)}} \leqslant k $ Here $ D_{T}={(t, x)\in \mathbb{R}^{2}:x\in [0, h(t))]} $, constants K and T only depend on $ h_{0} $, $ \alpha $ and $ \| u\|_{C^{2}([0, h_{0}])} $.

Proof Similar to the literature [7, 11, 12], the first step is to straighten the free boundary. This means that the free boundary problem (2.1) is reduced to a nonlinear initial boundary value problem on a fixed region through a function transformation. Let $ \xi(s) $ be a function in $ C^{3}[0, \infty] $ satisfying,

$ \begin{aligned} (t, s)\rightarrow (t, x) \ and\ x=s+\xi(s)(h(t)-h_{0}). \end{aligned} $

Consider the transformation, it changes the boundary $ x=h(t) $ to the line $ s=h_{0} $,

$ \begin{aligned} \dfrac{\partial s}{\partial x}=\dfrac{1}{1+\xi '(s)(h(t)-h_{0})}\equiv \sqrt{A(h(t), s)} \end{aligned} $

$ \begin{aligned} \dfrac{\partial^{2} s}{\partial x^{2}}=\dfrac{\xi ''(s)(h(t)-h_{0})}{[{1+\xi '(s)(h(t)-h_{0})}]^{3}}\equiv {B(h(t), s)} \end{aligned} $

$ \begin{aligned} -\dfrac{1}{h'(t)}\dfrac{\partial s}{\partial t}=\dfrac{\xi (s)}{1+\xi '(s)(h(t)-h_{0})}\equiv {C(h(t), s)} \end{aligned} $

Assume $ u(t, x)=u(t, s+\xi (s)(h(t)-h_{0}))=\varphi (t, s) $ then

$ \begin{aligned} u_{t}=\varphi_{t}-h'(t)C(h(t), s)\varphi_{s}, u_{x}=\sqrt{A(h(t), s)}\varphi_{s}, \\ u_{xx}=\sqrt{A(h(t), s)}\varphi_{ss}+B(h(t), s)\varphi_{s}. \end{aligned} $

Then (2.1) becomes initial boundary value problem on fixed region

$ \begin{equation} \left\{\begin{matrix} {\varphi_{t}}-Ad{\varphi_{ss}}-(Bd+h'C)\varphi_{s}=\varphi(a-b\varphi-\int_{0}^{t} {\phi }(t-\tau)\varphi(\tau) {\rm d}\tau & t>0, 0<s<h_{0} \\ {\varphi_{s}}(t, 0)=\varphi (t, h_{0})=0 & t>0 \\ \varphi=0, {h}'(t)=-\mu \frac{\partial \varphi}{\partial s} & t>0, s=h_{0} \\ h(0)={{h}_{0}}, \varphi(0, x)={{\varphi}_{0}}(s):=u_{0}(s) & 0<s<{{h}_{0}} \\ \end{matrix} \right. \end{equation} $

(3.1)

Here $ A=A(h(t), s) $, $ B=B(h(t), s) $, $ C=C(h(t), s) $. Although the free boundary is transformed into a fixed line, the equation itself becomes more complex. The equivalent equations (3.1) still contain a free boundary.

Denote $ h_{1}=-\mu u'_{0}(h_{0}) $ and $ 0<T\leq \frac{h_{0}}{8(1+h_{1})} $, define $ \mathcal{D}_{T}=[0, T]\times[0, h_{0}] $, we introduce the following metric space:

$ \begin{aligned} \mathcal{D}_{1}= \left\lbrace \varphi\in C(\mathcal{D}_{T}):\varphi (0, s)=u_{0}(s), \|\varphi -u_{0}\|_{C(\Delta T)}\leq 1 \right\rbrace, \end{aligned} $

$ \begin{aligned} \mathcal{D}_{2}= \left\lbrace h\in C^{1}([0, T]):h(0)=h_{0}, h'(0)=h_{1}, \|h'-h_{1}\|_{C([0, T])}\leq 1 \right\rbrace. \end{aligned} $

$ \mathcal{D}:=\mathcal{D}_{1}\times\mathcal{D}_{2} $ is a complete metric space with the metric, define a metric on $ \mathcal{D} $ :

$ \begin{aligned} d((\phi_{1}, h_{1}), (\phi_{2}, h_{2}))=\|\phi_{1}-\phi_{2}\|_{C(\Delta T)}+\|h'_{1}-h'_{2}\|_{C([0, T])}, \end{aligned} $

for any $ (\phi, h)\in \mathcal{D}_{1}\times\mathcal{D}_{2} $, we have:

$ \begin{aligned} \left| h(t)-h(0)\right| \leq \left| \frac{h(t)-h_{0}}{t}\cdot T\right| \leq T\|h'\|_{L^{\infty}} <\frac{h_{0}}{8(1+\left| h_{1}\right| )}(\|h_{1}\|_{L^{\infty}}+1)\leq \frac{h_{0}}{8}. \end{aligned} $

then, due to $ h_{1}, h_{2}\in \mathcal{D}_{1}\times\mathcal{D}_{2} $, $ h_{1}(0)=h_{2}(0)=h_{0} $

$ \begin{aligned} \| h_{1}-h_{2}\|_{C[0, T]} \leq T \| h'_{1}-h'_{2}\|_{C[0, T]}. \end{aligned} $

Refer to [11] $ L^{P} $ theory and Sobolev embedding theorem. For any $ (\phi, h_{1})\in \mathcal{D}_{1}\times\mathcal{D}_{2} $, the following initial boundary value problems

$ \begin{equation} \left\{\begin{matrix} {\bar{\varphi_{t}}}-Ad\bar{\varphi_{ss}}-(Bd+h'C)\bar{\varphi_{s}}=\varphi(a-b\varphi-\int_{0}^{t} {\phi }\varphi(t-\tau)\varphi(\tau){ {\rm d}}\tau) & t>0, 0<s<h_{0} \\ \bar{\varphi} (t, h_{0})=0, \frac{\partial \bar{\varphi}}{\partial s}(t, 0)=0 & t>0 \\ \bar{\varphi}(0, s)=u_{0}(s) & 0<s<{{h}_{0}} \\ \end{matrix} \right. \end{equation} $

(3.2)

have a unique solution $ \bar{\varphi}\in C^{(1+\alpha)/2, 1+\alpha}(\mathcal{D}) $, and $ \|\bar{\varphi}\|_{C^{(1+\alpha)/2, (1+\alpha)}(\mathcal{D}_{T})} \leq K_{1} $, here $ K_{1} $ is a constant depend on $ h_{0} $, $ \alpha $ and $ \|u_{0}\|_{C^{2}[0, h_{0}]} $, defining $ \bar{h}(t)=h_{0}-\int_{0}^{t}\mu \varphi_{s}(\tau, h_{0}){\rm d}\tau $ with $ \bar{h}'(t)=-\mu \phi_{s}(t, h_{0}) $, $ \bar{h}(0)=h_{0} $, $ \bar{h}'(0)=-\mu \phi_{s} (0, h_{0}) $ and hence $ \bar{h'}\in C^{\alpha /2}([0, T]) $, $ \|\bar{h'}\|_{C^{\alpha /2}([0, T])}\leq K_{2}:=\mu K_{1} $

Define $ \mathcal{F}:\mathcal{D}\rightarrow C([0, T]\times[0, h_{0}])\times C^{1}([0, T]) $ by $ \mathcal{F} (\phi, h)=(\bar{\phi}, \bar{h}) $ then

$ \begin{aligned} \| \bar{h'}-h_{0}\| _{C([0, h])}&\leq \| \bar{h'}\|_{C^{\alpha/2}([0, T])}T^{\alpha/2} \leq \mu K_{1}T^{\alpha/2}, \\ \| \bar{\varphi}-u_{0}\| _{C(\mathcal{D}_{T})}&\leq \| \bar{\varphi}-u_{0}\|_{C^{(1+\alpha)/2, 0}(\mathcal{D}_{T})}T^{(1+\alpha)/2} \leq K_{1}T^{(1+\alpha)/2}. \end{aligned} $

Obviously $ (\bar{\phi}, h)\in \mathcal{D} $ is a fixed point of the operator $ \mathcal{F} $ if and only if it is a solution to the problem (3.1). If we assume $ T\leq \min \left\lbrace (\mu K_{1})^{-2/\alpha}, K^{-2/(1+\alpha)}_{1}\right\rbrace $, then $ \mathcal{F} $ maps $ \mathcal{D} $ into itself.

$ \begin{aligned} \| \bar{\phi}_{1}-\bar{\phi}_{2}\|_{C^{(1+\alpha)/2, 1+\alpha}(\mathcal{D})} \leq K_{3} (\|\phi_{1}-\phi_{2}\|_{C(\mathcal{D}_{T})}+ \|h_{1}-h_{2}\|_{C^{1}[0, T]}). \end{aligned} $

Here $ K_{3} $ is depend on $ K_{1} $, $ K_{2} $. When $ T\leq 1 $, we have

$ \begin{aligned} \|\bar{\phi}_{1}-\bar{\phi}_{2}\|_{C^{(1+\alpha)/2, 1+\alpha}(\mathcal{D}_{T})}+\|\bar{h}'_{1}-\bar{h}'_{2}\|_{C^{\alpha/2}[0, T]} \leq K_{4} (\|\phi_{1}-\phi_{2}\|_{C(\mathcal{D}_{T})}+ \|h'_{1}-h'_{2}\|_{C[0, T]}). \end{aligned} $

Here $ K_{4} $ is depend on $ K_{3} $ and $ \mu $, $ T:=\min\left\lbrace 1, (\frac{1}{2K_{4}})^{2/\alpha}, (\mu K_{1})^{-2/\alpha}, K^{-2/(1+\alpha)}_{1}, \frac{h_{0}}{8(1+h_{1})}\right\rbrace $

$ \begin{aligned} \|\bar{\phi}_{1}-\bar{\phi}_{2}\|_{C(\mathcal{D}_{T})}+\|\bar{h}'_{1}-\bar{h}'_{2}\|_{C([0, T])} &\leq T^{(1+\alpha)/2}\|\bar{\phi}_{1}-\bar{\phi}_{2}\|_{C^{(1+\alpha)/2, 1+\alpha}(\mathcal{D}_{T})}+T^{\alpha/2}\|\bar{h}'_{1}-\bar{h}'_{2}\|_{C^{\alpha/2}([0, T])}\\ &\leq K_{4}T^{\alpha/2}(\|\phi_{1}-\phi_{2}\|_{C(\mathcal{D}_{T})}+\|h'_{1}+h'_{2}\|_{C([0, T])})\\ &\leq \frac{1}{2}(\|\phi_{1}-\phi_{2}\|_{C(\mathcal{D}_{T})}+\|h'_{1}+h'_{2}\|_{C([0, T])}). \end{aligned} $

Therefore, the operator $ \mathcal{F} $ is a compressed map acting on space $ \mathcal{D} $ for time $ T $ and it has a unique fixed point $ (\phi, h) $ on space $ \mathcal{D} $. In addition, using Schauder estimate, $ u\in C^{1+\alpha/2, 2+\alpha}((0, T]\times[0, h_{0}]) $, $ h(t)\in C^{1+\alpha/2}((0, T]) $. $ (u(t, x), h(t)) $ is the only locally classical solution to problem (3.1).

Lemma 3.2 (Comparison principle) Let $ \bar{h}\in C^{1}([0, +\infty)) $ and $ \bar{h}(t)>0 $ for $ t>0 $, condition (2.3) is satisfied. Assume that $ \bar{u}\in C^{(1+\alpha)/2, 1+\alpha}(\mathcal{D}), (\bar{u}, \bar{h}) $ satisfies

$ \begin{equation} \left\{\begin{matrix} {\bar{u}_{t}}\geq d{\bar{u}_{xx}}+\bar{u}(a-b\bar{u}-({{\phi }^{*}}\bar{u})) & t>0, 0<x<\bar{h}(t) \\ {\bar{u}_{x}}(t, 0)\geq 0, \bar{u}(t, \bar{h}(t))=0 & t>0 \\ \bar{h}'(t)\geq -\mu {\bar{u}_{x}}(t, \bar{h}(t)) & t>0. \\ \end{matrix} \right. \end{equation} $

(3.3)

If $ \bar{h}(0)\geq h_{0} $, $ \bar{u}(0, x)\geq 0 $ in $ [0, \bar{h}(0)] $ and $ \bar{u}(0, x)\geq u_{0}(x) $ in $ [0, h_{0}] $. Then the solution $ (u, h) $ of (2.1) satisfies $ \bar{h}(t)>h(t) $ in $ [0, \infty) $ and $ u\leq \bar{u} $ in $ \mathcal{D} $.

Proof According to [11, 13], the comparison principle that $ u(t, x)\leq \bar{u}(t, x) $ for $ t\in (0, T_{0}) $ and $ x\in [0, h(t)) $, $ \bar{u}(t)=\frac{a}{b}e^{\frac{a}{b}t}(e^{\frac{a}{b}t}-1+\frac{a}{b\|u_{0}\|_{\infty}})^{-1} $ is the solution of the problem: $ \frac{{{\rm d}} \bar{u}}{{{\rm d}}{t}}=\bar{u}(a-b\bar{u}) $, $ t>0 $, $ \bar{u}(0)=\|u_{0}\|_{\infty} $.

Then we get $ u(t, x)\leq \sup_{t\geq 0}{\bar{u}(t)}=\max\left\lbrace \frac{a}{b}, \|u_{0}\|_{\infty}\right\rbrace:=K_{1} $ with $ (t, x)\in (0, T_{0}\times(0, h(t)] $. Follows from strong maximum principle and Hopf lemma that $ u(t, x)>0 $, $ u_{x}(t, h(t))<0 $ with $ (t, x)\in (0, T)\times(0, h(t)] $. When $ t\in (0, T_{0}) $, given by Stefan condition $ h'(t)>0 $, we define:

$ \begin{aligned} \Omega_{K}^{h}:&=\left\lbrace (t, x)\in \mathbb{R}^{2}:0<t<T_{0}, h(t)-K^{-1}<x<h(t)\right\rbrace \\ \bar{u}&(t, x)=K_{1}[2K(h(t)-x)-K^{2}(h(t)-x)^{2}], \end{aligned} $

here $ \bar{u}(t, x) $ is an auxiliary function, $ K $ is a given positive constant. When $ (t, x)\in \Omega_{K}^{h} $

$ \begin{aligned} \bar{u}(t, x)=2KK_{1}[1-K(h(t)-x)]h'(t)\geq 0, -\Delta\bar{u}=2K^{2}K_{1}, \\ {u}(a-b{u}-({{\phi }^{*}}{u}))\leq aK_{1}\ and\ \bar{u}_{t}-d\Delta \bar{u}\geq 2dK^{2}K_{1}\geq aK_{1}. \end{aligned} $

If $ K\geq \sqrt{\frac{a}{2d}} $, $ u(t, h(t)-K^{-1})\leq \bar{u}(t, h(t)-K^{-1}) $ and $ u(t, h(t))=\bar{u}(t, h(t))=0 $. Besides, when $ x\in [h_{0}-K^{-1}, h_{0}] $, we have

$ \begin{aligned} \bar{u}(0, x)=K_{1}[2K(h_{0}-K^{2}(h_{0}-x)^{2})]\geq KK_{1}(h_{0}-x)\\ u_{0}(x)=-\int_{x}^{h_{0}}u'_{0}(s){{\rm d}}s \leq (h_{0}-x)\|u'_{0}\|_{C([0, h_{0}])}. \end{aligned} $

According to the maximum principle and $ u(t, x)\leq \bar{u}(t, x) $ with $ (t, x)\in\Omega_{K}^{h} $, we can obtain

$ \begin{aligned} h'(t)=-\mu u_{x}(t, h(t))\leq -\mu\bar{u}_{x}(t, h(t))=2\mu KK_{1}:=K_{2}. \end{aligned} $

Therefore, existence constant $ K_{2}>0 $ that does not depend on $ T_{0} $. The proof is complete.

Red fire ants (scientific name: Solenopsis invicta Buren) are considered one of the 100 most dangerous invasive organisms globally. Studying the diffusion law of red fire ants is crucial to prevent their large-scale spread. Transmission of red fire ants occurs naturally or through human activity. Natural transmission mainly occurs through the propagation of reproductive ants by flying or during flood flow. It can also occur when nests are moved over short distances. Artificial transmission through human activities can occur over longer distances.

Based on research from [16-18], the red fire ant primarily spreads naturally in limited spaces. Using data from the aforementioned paper, we simulated the initial propagation of the red fire ant invasion, with less intermediate competition. In the absence of any influence from other species, the early spread of the red fire ant invasion was simulated. Here $ u $ represents ant nest distribution area (unit $ m^{2} $), $ t $ represents propagation time (unit year), $ x $ is the spread distance (unit $ km $). The following table lists the parameters:

The equation was discretized using the finite difference method, a common numerical solution for partial differential equations. This method transforms continuous partial differential equations into discrete difference equations by approaching the differential operator, and obtains a numerical solution by solving the difference equation. The accompanying figure was generated using Matlab calculations. Figure 1 and Figure 2 show the distribution area and number of ant nests, respectively. The results indicate a slow increase in both the distribution area and number of ant nests in the early stage, followed by an acceleration with the increase of invasion time.

Error analysis is a crucial aspect of numerical solutions. Errors primarily arise from insufficient precision when using the finite difference method to approximate the analytical solution or from local truncation errors when approaching the free boundary. Additionally, rounding errors can occur when values are rounded during numerical calculations on a computer. In this paper, the time step is taken as $ h_{t}=0.005 $, the space step is taken as $ h_{x}=0.05 $. Based on the experimental data, the error analysis of the early population expansion of red fire ant was carried out and the results were analyzed by error and root-mean-square error (RMSE).

The error is calculated as the average of the absolute difference between the actual value and the experimental value. This method provides an intuitive way to observe the difference between the experiment and the actual value. The root-mean-square error, also known as the standard error, is the square root of the ratio of the sum of the squared deviations from the observed value to the number of observations (n). This measure is highly sensitive to large or small errors in a set of measurements. It is evident that the experimental data is highly accurate.

The simulation data indicates that the distribution area of ant nests significantly increased when the annual transmission distance exceeded 7 km in a limited area, due to the limited autonomous transmission ability of red fire ants. The fastest transmission time was 8 km, with relatively small rest time. The number of ant nests reached its maximum when the annual dispersal distance was 8 km. Therefore, the focus of the prevention and control of red fire ants should be to strictly control the transport of goods in the red fire ant occurrence area to prevent the spread caused by human carrying.