Fig. 1 Fitness landscape of phenotypes. The solid line shows the fitness landscape observed. When an organism evolves on such a multipeak fitness landscape, it’s easy to be trapped at a lower local peak. The dashed line displays the phenotypic fitness landscape smoothed by reaction norm, i.e., the expect fitness for a genotype with a certain average phenotype. This fitness landscape just has one single peak, which makes it easier for any phenotype to evolve to another one with higher fitness.

Fig. 2 An evolutionary process for phenotype in a population from x0 to xn. An individual selects the next phenotype to evolve with a certain probability according to its instantaneous fitness at each time step, thus many possible paths of phenotype in a population occur including paths a and b. Suppose that trajectory a consists of phenotype ${{x}_{0}},{{x}_{1}},{{x}_{2}},...,{{x}_{n}}$, then the possibility is $P[{{x}_{0}}({{t}_{0}}),{{x}_{1}}({{t}_{1}}),{{x}_{2}}({{t}_{2}}),\ldots,{{x}_{n}}({{t}_{n}})]$。

Fig. 3 The distribution of evolutionary paths. Different colors represent different distribution probability of paths. As time goes by, one phenotype differentiates to other phenotypes with different probabilities. The evolutionary path is continuous in this process because that phenotypes can only evolve to adjacent phenotypes. If the evolutionary results are observed at the moment of t = 1,000, significant discontinuous phenotype differentiation (blue regions) can be seen.

Fig. 4 Path-dependent evolutionary process of states. (a) The frequency distribution curve of states over time. The states represent genotype, phenotype or ecological process. (b) The frequency distribution curve of phenotypes at t = 10, (c) The frequency distribution curve of phenotypes at t = 600. From the perspective of phenotype, the distribution curve changes from unimodal in the early phase to bimodal at t = 600, which means the phenotype differentiates into two phenotypes (25 and 28) at t = 600 from one phenotype (50) in the initial stage. Then we can judge if the species with phenotype 25 and 28 are different species.

Fig. 5 Frequency distribution of phenotypes and genotypes at different time. (a) showing the frequency distribution curve of the state x1(x2) in the early differentiation. (b) demonstrating the frequency distribution curve of the phenotype x1 at a certain moment after differentiation. (c) indicating the frequency distribution curve of the genotype x2 at the same moment after differentiation. Suppose that any two states x1 and x2 of the original species A evolved influenced by random mutation and drift, then different degree of differentiation occurs in the same or different time. When observing in a “cutting plane” of a species evolution, we notice that x1(x2) differentiates into x11(x21) and x12(x22), and the evolution are path-dependent. Meanwhile, if the differentiated phenotypes and the genotypes in a quantity statistics satisfy certain conditions, the individuals with ${{x}_{11}},{{x}_{21}}({{x}_{11}},{{x}_{22}},or{{x}_{12}},{{x}_{21}},or{{x}_{12}},{{x}_{22}})$ are new species, respectively.

Fig. 6 schematic diagram of the species delimitation over time. The state x1 (x2) differentiates at t1(t2). It can be determined whether a specie at different moments is a new species.

Fig. 7 Differentiation of three independent phenotypes from one phenotype of a species at a moment. The figure is a special case of judging whether the phenotypic differentiates completely. There is a continuous and a little difference for the phenotype ?1 within the species (the difference range is ± 0.15), and the phenotype 0 versa. The discontinuity of these two interspecies phenotypes is 1, and the discontinuity is much higher than the intraspecies continuous difference. Therefore, it can be considered that the phenotype meets the requirements for phenotype in the concept of morphological species. Taking into account the differentiation of another state of this species, if the difference satisfies the same conditions, the corresponding individuals can be determined as a new species.

Fig. 8 Different case of phenotype differentiation. Imagine the relation of difference has been satisfied for one biological character. Then we consider phenotypes. In figure 8(a), $d=|{{x}_{1}}-{{x}_{2}}|=|(-6)-6|=12,{{\sigma }_{1}}=3,{{\sigma }_{2}}=3.5,d>{{\sigma }_{1}}$ and $d>{{\sigma }_{2}}$. If individuals in two populations satisfy these conditions, they are different species. In figure 8(b), $d=|{{x}_{1}}-{{x}_{2}}|=|(-2)-2|=4,{{\sigma }_{1}}=4,{{\sigma }_{2}}=3.5,d={{\sigma }_{1}}$ and $d>{{\sigma }_{2}}$. If those individuals satisfy these conditions, they are the same species. In figure 8(c), $d=|{{x}_{1}}-{{x}_{2}}|=|(-2)-2|=4,{{\sigma }_{1}}=8.1,{{\sigma }_{2}}=8.2,d<{{\sigma }_{1}}$ and $d<{{\sigma }_{2}}$. If those individuals satisfy these conditions, they are the same species.