Seeds of lettuce (Lactuca sativa) and beans (Phaseolus vulgaris) were purchased from a commercial store in Buea, Cameroon. Seeds were sterilised in 2.0% sodium hypochloride (NaOCl) for two minutes and rinsed with distilled water before being used in each of the experiments [18] . Preliminary germination tests using the blotter method (method similar to control with distilled water) were performed on the seeds to ascertain viability.

Five invasive species were selected from agricultural fields around Mount Cameroon for their significant ecological and agricultural impacts. They serve as a macrocosm of the broader challenges posed by invasive plants in the region, serving as a basis for further research and action to mitigate their effects on both agriculture and the environment.

For preparation of fresh leaves aqueous and leachates extracts, Chromoleana odorata, Ageratum conyzoides, Bidens pilosa, Tithonia diversifolia and Cecropia peltata were collected from the Mount Cameroon Region latitude 3057'N to 4028'N and longitude 8058'E to 9204'E [19] . The plants were transported to the Life Science Laboratory of the Faculty of Science and the Teaching and Research Farm of the Faculty of Agriculture and Veterinary Medicine of the University of Buea. The leaves of each plant collected were separated for use in the bioassays. The leaves were washed with tap water, then with distilled water, and allowed to dry for 2 hours before extraction.

For fresh aqueous extracts of fresh plant leaves, a litre of distilled water was added to two hundred grams of chopped fresh leaves of each invasive alien species in a blender and homogenized for 5 minutes at room temperature. The mixture was allowed to stand for 30 minutes before filtration through Whatman No. 1 filter paper.

For foliage leachate extracts of fresh plant leaves, 200 g of fresh leaves of each invasive alien species were immersed in 1 litre of distilled water for 5 minutes and then leached water was collected and filtered through a Whatman No.1 filter paper. The filtrates were considered as full-strength concentrations and were stored at 4˚C until used. The stock solution of each plant was diluted in distilled water to obtain 100mL of the working solutions (0%, 25%, 50%, 75%, and 100%).

With respect to the seed bioassay, specific concentrations of 25%, 50%, 75%, and 100% were used for the extracts and leachates. These concentrations were chosen to reflect a gradient that allows for the assessment of the varying effects of the extracts on target plants. Ecologically, these concentrations are relevant as they simulate different scenarios in natural settings where allelochemicals may be present in varying amounts due to environmental conditions, plant density, and decomposition rates.

For bioassay with fresh aqueous extract, ten (10) seeds each of Lactuca sativa and Phaseolus vulgaris were placed separately in each of three sterilised (9 cm diameter) Petri dishes lined with two layers of Whatman No.1 moistened filter paper. 10 mL of solution containing 25%, 50%, 75% and 100% full-strength extract diluted with distilled water was added to the dishes. For the control treatment, 10 mL of distilled water (0%) was added per petri dish. The dishes were incubated in the dark for one week and data on germination, root length and shoot length were recorded.

For bioassay with fresh foliage leachate, ten (10) seeds each of Lactuca sativa and Phaseolus vulgaris were placed separately in each of three sterilised (9 cm diameter) Petri dishes lined with two layers of Whatman No.1 moistened filter paper. 10 mL of solution containing 25%, 50%, 75% and 100% full-strength extract diluted with distilled water was added to the dishes. For the control treatment, 10 mL of distilled water (0%) was added per petri dish. The dishes were incubated in the dark for one (1) week and a germination test was carried out under the condition of a 12 h light/dark cycle with 25 ± 2 temperature. Data on germination, root length and shoot length where germination for each seed was recorded once its radicle protruded to ≥1 mm in length. Lettuce was recorded as (SP1) and beans as (SP2). The percentage of seed germination was calculated as in Eq. (1):

$\text{Germination percentage}\left(\text{GP}\right)=\frac{\text{Cumulative number of germinated seed}}{\text{Total number of seeds}}*100$ (1)

The root and shoot lengths were obtained by separately measuring the root and shoot lengths of the samples from each treatment by tracing with an inelastic string and reading the length of a meter rule.

A piece of land measuring 100 m × 100 m was prepared and divided equitably into 15 mini-plots. Fourteen of the minis-plots were selected for the treatments and the remaining was considered as control and respectively labelled from 1 - 15. Rhizospheric soil samples were collected from natural stands dominated by Ageratum conyzoides, crushed into fine powder and weighed into 0.5 Kg, 1 Kg, 1.5 Kg, 2 Kg, 2.5 Kg, 3 Kg, 3.5 Kg, 4 Kg, 4.5 Kg, 5 Kg, 5.5 Kg, 6 Kg, 6.5 Kg and 7 Kg packs. Bean seeds were obtained from a local market in Buea. The control setup involved using soil collected from an area dominated by non-invasive species, specifically not occupied by Ageratum conyzoides. This soil was then crushed into a fine powder and weighed, mirroring the preparation method used for the treatment plots.

Starting with the mini-plots marked (1), 60 bean seeds were sown on each of the three sections after which 0.5 Kg of rhizospheric soil sample was fully spread, by broadcasting on each plot. The method was repeated using the remaining mini-plots (2)-(14) with respectively 1 Kg, 1.5 Kg, 2 Kg, 2.5 Kg, 3 Kg, 3.5 Kg, 4 Kg, 4.5 Kg, 5 Kg, 5.5 Kg, 6 Kg, 6.5 Kg and 7 Kg of the samples. 60 seeds were also planted on each section of mini-plots, including mini-plots (15), which served as the control.

The setup was watered twice daily with water until the beans fully germinated within 9 days.

Each experimental setup was repeated three times. Data were tested for normality and homogeneity of variance. This was followed by the analysis of variance through the GLM approach, with natural log transformation for data that did not meet the conditions of normality and homoscedasticity. Where significant differences exist, means were separated through the Tukey HSD test at α = 0.05.

Data collection and analysis of the polynomial regression model

Data on the total number of germinated seeds per plot was collected and based on the data a mathematical model was formulated using polynomial regression according to [10] . All data in the modelling process was analysed using Scilab 6.1.0 computer software with its Curve Fitting Toolbox.

Polynomial regression is a special case of multiple regressions, with only one independent variable x. The kth order polynomial regression model in one variable can be expressed in Eq. (2):

$y={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2+{\beta }_3{x}^2+\dots +{\beta }_k{x}^k+ϵ$ (2)

where k is the degree of the polynomial also known as the order of the model.

The mean squared error MSE is an unbiased estimator of the variance ${\sigma }^2$ of the random error term Eq. (3):

$MSE=\frac{{{\displaystyle \sum }}_{i=1}^n{(y_i-{\hat{y}}_i)}^2}{n-\left(k+1\right)}$ (3)

where, $y_i$ are observed values, ${\hat{y}}_i$ are the fitted values of the dependent variable Y for the ith case and $n-\left(k+1\right)$ is the degree of freedom. Since the mean squared error is the average squared error, where averaging is done by dividing by the degrees of freedom, MSE is a measure of how well the regression fits the data. The root mean squared error, RMSE is given by the square root of the mean square error.

RMSE = $\sqrt{MSE}$.

The R-squared $R^2$ (coefficient of determination) of the regression equation (Eq. 3) is defined as

$R^2=1-\frac{{{\displaystyle \sum }}_{i=1}^n{(y_i-{\hat{y}}_i)}^2}{{{\displaystyle \sum }}_{i=1}^n{(y_i-{\bar{y}}_i)}^2}$ (3)

where, $\bar{y}$ is the arithmetic mean of the y variable. $R^2$, measures the percentage of variation in the response variable y explained by the explanatory variable x. The value of $R^2$ranges between zero and one ( $0\le R^2\le 1$). An $R^2$ value of 0.9 or above is very good, but when the $R^2$ value is 0.5 or below, the regression explains only 50 % or less of the variation in the data, therefore giving a poor prediction [11] .

Adjusted R-squared $R^{*}{}^2$ (Eq. 4) is computed as follows

$R^{*}{}^2=R^2-\frac{\left(1-R^2\right)k}{n-\left(k+1\right)}$ (4)

The result in Figure 1 shows that invasive weed species leaf extract at different concentrations significantly (P < 0.05) influenced the root length of lettuce (SP1) and beans (SP2). The result revealed a consistent decrease in root length of lettuce (SP1) and beans (SP2) with increasing concentrations of leaf extract across all invasive weed species, with control having the highest root length (5.2 cm) and 100% concentration the lowest root length (0.4 cm) across ( Figure 1 ).

A similar result was observed for shoot length where increasing invasive weed species concentrations significantly reduced the shoot length of both lettuce (SP1) and beans (SP2) ( Figure 2 ). The result also showed that lettuce (SP1) shoot length was more significantly reduced by increasing weed species concentration than beans (SP2) shoot length ( Figure 2 ). Shoots of beans (SP2) was longer (6.2 cm) at Bidens pilosa 25% concentration while lettuce (SP1) shoot length was highest (3.9 cm) at the control ( Figure 2 ).

The results in Figure 3 show that the germination rate of SP2 was higher for all treatments with a 100% germination rate in the control. The germination was significantly hindered by Tithonia diversifolia treatment at all concentrations.

The result shows the allelopathic potential of fresh leaf leachates on the root length of SP1 and SP2 with a significant decrease of root length as leaf leachate concentration ( Figure 4 ). Control plants with no allelopathic potential invasive weed species had the longest root lengths for SP1 and SP2 (2.37 cm and 5.87 cm respectively) with the shortest root lengths in plants treated with leaf leachate of Ageratum conyzoides at 100% concentrations for SP1 (0.1cm) and SP2 (0.2 cm).

The pattern of variation of the shoot length in SP1 and SP2 is observed in Figure 5 . Plants with the longest shoots for SP1 were those in the Cecropia peltata treatment at 25% concentration (3.8 cm) while SP2 plants with the longest shoots were those under the control treatment (7.63 cm). SP1 plants with the shortest shoots were those that received leachate from Ageratum conyzoides at 100 concentrations (0.1 cm) and at 100% concentration for SP2 (0.4 cm) ( Figure 5 ).

The pattern of variation of germination percentages in SP1 and SP2 is shown in Figure 6 . Plants with the highest germination percentage for both SP1 and SP2 were those in the control treatment (85.1% and 100% respectively). Plants treated with leachate extracts at various concentrations experience a significant reduction in the germination rate with 100% concentration of Tithonia diversifolia leachate exerting the most effect (15.1%).

The results show that as the quantity of rhizospheric soil increased, the percentage of germination of the bean seeds decreased ( Table 1 ).

The statistical outputs of the polynomial regression model are given in Table 2 .

The best fit polynomial regression model occurred at polynomial of degree 8 and its parameter estimates are seen in Equation (5).

$\begin{array}{c}y=100.01397-28.933788x+74.985596x^2-80.294922x^2+41.541115x^4\\ -11.747532x^5+1.8501702x^6-0.1519795x^7+0.0050631x^8\end{array}$ (5)

where x = concentrations of rhizospheric soil and y = number of germinated seeds.

Figures 7 and 8 show the comparisons of the polynomial models with the measured data. A curvilinear relationship was observed and polynomials of degrees 2, 3, 4, 5, 6, 7, 8 and 9 (1 to 9) were fitted as shown in Figure 7 and 8 respectively. The best fit polynomial regression model occurred at a polynomial of degree 8 (8) ( Figure 8 ).

The results of this study showed that different strengths of leaf extracts and leaf leachates of the selected plants significantly inhibited seed germination and seedling growth of lettuce and beans. The observed inhibition of the germination of lettuce and bean seeds could be attributed to a contribution of allelochemicals present in the leaf extracts of Ageratum conyzoides, Bidens pilosa, Cecropia peltata, Tithonia diversifolia and Chromoleana odorata. These allelochemicals have various chemical structures and phytotoxic effects responsible for inhibiting seed germination and plant growth via interferences with core biochemical and physiological processes such as cell division, energy transfer, water and nutrient uptake [20] . Different studies have shown that allelochemicals are water-soluble and can accumulate upon release within seeds in direct contact with bioactive concentrations [21] . Therefore, the inhibitory effect may be due to the entry of water-soluble allelochemicals into the seed supporting this study [22] . Similar results were obtained by [23] who investigated the allelopathic activity of fresh leaf extracts and leachates of Diplotaxis erucoides on four crops. These results also corroborate the findings of [24] [25] who reported the allelopathic effects of extracts of different weeds on seedling growth and germination of test plants.

The inhibition was observed as concentration-dependent. The germination percentage of seeds treated with 100% and 75% weed extracts was significantly lower than those treated with 50% and 25% extracts, respectively. Pronounced increases in allelopathic inhibition due to the effect of dosage have been reported [26] . Fresh leaf extract and leaf leachates exerted some stimulatory effects at the lowest applied concentrations (25%) in some germination parameters. The observed stimulation could be due to growth-promoting substances in tissues [27] . This phenomenon is called hormesis [28] and has been recorded in several studies in allelopathy. Germination was observed to be higher in the control than in the applied extract treatments. This may be due to the absence of leaf extract concentration facilitating water uptake which is an important component in stimulating some metabolic and physiological processes occurring in seed germination [29] .

The changes in the root and shoot length reflect the allelopathic influences on each organ. The results of this study indicated that the extracts of the selected weeds inhibited root and shoot growth of lettuce and bean seeds and it was observed that the roots were more sensitive to the extracts than the shoots. This shows an organ-based sensitivity of the species to phytotoxic compounds. Possible reasons are that roots are the first to emerge and are in direct contact with extracts and thus are exposed to peak periods and concentrations of phytotoxins. According to [30] , the highest level of allopathic suppression occurs when maximum levels of phytotoxins coincide with the early stages of plant growth. Similar findings were reported by [31] who investigated the allelopathic potentiality of Euphorbia hypericifolia L. on germination and seedling development of sympatric crops and weeds.

Fresh leaf extracts showed more inhibition than the leachates and this may be because the chemicals were released slowly in the leachates. The most effective allelopathic weed was Ageratum conyzoides followed by Tithonia diversifolia while Bidens pilosa had the least inhibitory effect on lettuce and bean growth. Ageratum conyzoides exhibits stronger allelopathic effects due to its potent secondary metabolites, especially volatile oils and water-soluble phytochemicals, which inhibit the growth of various crops. Its adaptability to diverse environments and invasive nature further enhance its competitive edge. Research comparing its chemical compositions with other species can help to identify specific allelochemicals responsible for these effects, highlighting the importance of chemical type and concentration in allelopathy. However, the chemical composition was not compared between species in this study.

Rhizospheric soil samples of Ageratum conyzoides were used to explore the effects of the extracts on the germination and growth of bean seeds. The addition of the samples of the rhizospheric soil affected the germination rate of bean seeds implying that the germination rate was dependent on the quantity of rhizospheric soil used for the treatments. Similar results were reported by [7] who tested the effects of rhizospheric soil of Tectona grandis L plantation on the germination of Lycopersicum esculentum. Environmental variables like soil composition and climate significantly influence the allelopathic effects of plants in natural settings compared to controlled conditions. Soil nutrients, pH, and microbial communities can affect the availability and effectiveness of allelochemicals, while climate factors such as temperature and humidity impact their production and release. In controlled environments, these variables are standardized, which may not reflect the complex interactions present in natural ecosystems, leading to different allelopathic outcomes.

The purpose of our study was to model the allelochemical effect of the rhizospheric soil sample of Ageratum conyzoides on the seed germination of Phaseolus vulgaris (bean seeds). From our results obtained from the field experiment, a scattered plot showed a curvilinear relationship between the rhizospheric soil treatments and the mean number of germinated seeds and polynomials of degrees 2, 3, 4, 5, 6, 7, 8 and 9 were fitted to see which of the models will provide a good approximation of the relationship that exist between plotted data points. The best fit was observed at polynomial of degree 8 with the lowest error statistic (RMSE = 1.2961592) and high deterministic coefficient (R² = 0.9985126) (Table 2). The value of R² at best fit polynomial degree 8 denotes that the allelopathic concentration of rhizospheric soil of Ageratum conyzoides is responsible for 99.85 % variability of germination percentage in Phaseolus vulgaris and 0.15% (100% - 99.85%) of the variation is caused by other factors. Though R-squared (R²) indicates how well a regression model fits a set of data, the “Adjusted R Square” (R*²) is preferred in the interpretation of results to avoid false relationships and should always be less than or equal to R Square with small differences between them for the best fit of a model [9] . [10] states that, R² shows how well data points fit a regression line assuming every single variable explains the variation in the dependent variable which is not true. Whereas, adjusted R Square tells how well the data points fit a regression line showing the percentage of variation explained only by the independent variables that affect the dependent variable. From the results, the value of the adjusted R square (R*² = 0.9965294) for our best fit polynomial model of degree 8 indicates that truly, 99.65% of the variation in the germination percentage in Phaseolus vulgaris is caused by rhizospheric soil of Ageratum conyzoides and the remaining 0.35% of the variation is caused by other factors.

The standard error is a measure of the precision of the model and indicates how wrong the estimated coefficient could be if used to make predictions. It should be as small as possible relative to the estimated coefficient. The error statistic (RMSE) of the best fit of the model (polynomial of degree 8) in this study is 1.2961592 indicating that on average, our estimates of germination percentage with this model will be wrong at 1.29. The cubic spline method was used to smoothen the model and reduce error due to the Runge’s phenomenon (problem of oscillations at the edges of an interval that occurs when using the polynomial of a high degree over a set of equally spaced interpolation points).